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\title{
\rule{\linewidth}{1pt} \\
\textbf{
    Algebraic Groups
  }
    \\ {\normalsize University of Georgia, Fall 2020} \\
  \rule{\linewidth}{2pt}
}
\titlehead{
    \begin{center}
  \includegraphics[width=\linewidth,height=0.45\textheight,keepaspectratio]{figures/cover.png}
  \end{center}
       \begin{minipage}{.35\linewidth}
    \begin{flushleft}
      \vspace{2em}
      {\fontsize{6pt}{2pt} \textit{Notes: These are notes live-tex'd
from a graduate course in Algebraic Groups taught by Dan Nakano at the
University of Georgia in Fall 2020. As such, any errors or inaccuracies
are almost certainly my own. } } \\
    \end{flushleft}
    \end{minipage}
    \hfill
    \begin{minipage}{.65\linewidth}
    \end{minipage}
  }







\begin{document}

\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2021-07-16 }
\end{flushleft}


\newpage

% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage

\hypertarget{prologue}{%
\section*{Prologue}\label{prologue}}
\addcontentsline{toc}{section}{Prologue}

\hypertarget{references}{%
\subsection{References}\label{references}}

\begin{itemize}
\item
  Carter's ``Finite Groups of Lie Type''\autocite{carter_1985}
\item
  Humphreys' ``Linear Algebraic Groups''\autocite{humphreys_2004}
\item
  Course Videos:
  \url{https://www.youtube.com/playlist?list=PLYKpGBt1pueCLzs_b-VYKe_bX9My-3nUt}
\end{itemize}

\hypertarget{notation}{%
\subsection{Notation}\label{notation}}

\todo[inline]{Todo}

\newpage

\hypertarget{outline}{%
\subsection{Outline}\label{outline}}

Guide for Math 8190 Videos:

\begin{align*}9-14-20\end{align*}
A survey on the representation theory of finite-dimensional algebras

\begin{align*}9-25-20\end{align*}
Frobenius kernels, simple G\_r-modules, Steinberg's Tensor Product
Theorem, Connections with Lusztig's Conjecture

\begin{align*}9-28-20\end{align*}
Kempf's Vanishing Theorem, Good and Weyl filtrations, Cohomology

\begin{align*}10-2-20\end{align*}
Good and Weyl Filtrations

\begin{align*}10-5-20\end{align*}
Highest Weight Categories

\begin{align*}10-7-20\end{align*}
Examples: Schur algebras, tilting modules, Weyl modules, Bott-Borel-Weil
Theorem

\begin{align*}10-9-20\end{align*}
Bott-Borel-Weil Theory

\begin{align*}10-12-20\end{align*}
Bott-Borel-Weil Theorem with applications

\begin{align*}10-14-20\end{align*}
Basic Properties of Characters, Weyl's Character Formula

\begin{align*}10-16-20\end{align*}
Weyl's Character Formula

\begin{align*}10-19-20\end{align*}
Linkage, Strong Linkage, Examples

\begin{align*}10-21-20\end{align*}
Strong Linkage Principle, Translation Functors (intro)

\begin{align*}10-23-20\end{align*}
Translation Functors

\begin{align*}10-26-20\end{align*}
Translation Functors and Characters

\begin{align*}10-28-20\end{align*}
Properties of Translation Functors

\begin{align*}11-2-20\end{align*}
Translation, Wall Crossing, Lusztig's Conjecture

\begin{align*}11-4-20\end{align*}
Representations of G\_rT and G\_rB

\begin{align*}11-6-20\end{align*}
Representations of G\_rT and G\_rB, Good (p,r)-filtrations, Donkin's
(p,r)-Filtration Conjecture

\begin{align*}11-9-20\end{align*}
Strong Linkage for G\_rT-modules, Extensions for G\_rT-modules,
Steinberg Modules

\begin{align*}11-11-20\end{align*}
Filtrations and Reciprocity for G\_rT and G\_r Modules

\begin{align*}11-13-20\end{align*}
G\_rT-modules and reciprocity, Injective G\_rT-modules, Humphreys-Verma
Conjecture, Donkin's Tilting Module Conjecture

\begin{align*}11-16-20\end{align*}
G-structures for injective G\_r-modules, Donkin's Tilting Module
Conjecture

\begin{align*}11-18-20\end{align*}
Injective G\_T-modules and Injective G-modules

\begin{align*}11-20-20\end{align*}
Cohomology of Frobenius kernels, finite generation of cohomology

\begin{align*}11-23-20\end{align*}
Cohomology of Frobenius kernels, restricted nullcone, Jantzen's
Conjecture on support varieties, conjectures for low primes

\hypertarget{friday-august-21}{%
\section{Friday, August 21}\label{friday-august-21}}

\hypertarget{intro-and-definitions}{%
\subsection{Intro and Definitions}\label{intro-and-definitions}}

\begin{definition}[Affine Variety]

Let \(k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
be algebraically closed
(e.g.~\(k = {\mathbb{C}}, \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\)).
A variety \(V\subseteq k^n\) is an \emph{affine \(k{\hbox{-}}\)variety}
iff \(V\) is the zero set of a collection of polynomials in
\(k[x_1, \cdots, x_n]\).

\end{definition}

Here \({\mathbb{A}}^n\coloneqq k^n\) with the Zariski topology, so the
closed sets are varieties.

\begin{definition}[Affine Algebraic Group]

An \emph{affine algebraic \(k{\hbox{-}}\)group} is an affine variety
with the structure of a group, where the multiplication and inversion
maps
\begin{align*}  
\mu: G\times G &\to G \\
\iota: G&\to G
\end{align*}
are continuous.

\end{definition}

\begin{example}

\(G = {\mathbb{G}}_a \subseteq k\) the \emph{additive group} of \(k\) is
defined as \({\mathbb{G}}_a \coloneqq(k, +)\). We then have a
\emph{coordinate ring} \(k[{\mathbb{G}}_a] = k[x] / I = k[x]\).

\end{example}

\begin{example}

\(G = \operatorname{GL}(n, k)\), which has coordinate ring
\(k[x_{ij}, T] / \left\langle{\operatorname{det}(x_{ij})\cdot T = 1}\right\rangle\).

\end{example}

\begin{example}

Setting \(n=1\) above, we have
\({\mathbb{G}}_m \coloneqq\operatorname{GL}(1, k) = (k^{\times}, \cdot)\).
Here the coordinate ring is
\(k[x, T] / \left\langle{xT = 1}\right\rangle\).

\end{example}

\begin{example}

\(G = {\operatorname{SL}}(n, k) \leq \operatorname{GL}(n, k)\), which
has coordinate ring
\(k[G] = k[x_{ij}] / \left\langle{\operatorname{det}(x_{ij}) = 1}\right\rangle\).

\end{example}

\begin{definition}[Irreducible]

A variety \(V\) is \emph{irreducible} iff \(V\) can not be written as
\(V = \cup_{i=1}^n V_i\) with each \(V_i \subseteq V\) a proper
subvariety.

\begin{figure}
\centering
\includegraphics{figures/Reducible_v_irreducible.png}
\caption{Reducible vs Irreducible}
\end{figure}

\end{definition}

\begin{proposition}[?]

There exists a unique irreducible component of \(G\) containing the
identity \(e\). Notation: \(G^0\).

\end{proposition}

\begin{proposition}[?]

\(G\) is the union of translates of \(G^0\), i.e.~there is a
decomposition
\begin{align*}  
G = {\textstyle\coprod}_{g\in \Gamma} \, g\cdot G^0
,\end{align*}
where we let \(G\) act on itself by left-translation and define
\(\Gamma\) to be a set of representatives of distinct orbits.

\end{proposition}

\begin{proposition}[?]

One can define solvable and nilpotent algebraic groups in the same way
as they are defined for finite groups, i.e.~as having a terminating
derived or lower central series respectively.

\end{proposition}

\hypertarget{jordan-chevalley-decomposition}{%
\subsection{Jordan-Chevalley
Decomposition}\label{jordan-chevalley-decomposition}}

\begin{proposition}[Existence and Uniqueness of Radical]

There is a maximal connected normal solvable subgroup \(R(G)\), denoted
the \emph{radical of \(G\)}.

\begin{itemize}
\tightlist
\item
  \(\left\{{e}\right\} \subseteq R(G)\), so the radical exists.
\item
  If \(A, B \leq G\) are solvable then \(AB\) is again a solvable
  subgroup.
\end{itemize}

\end{proposition}

\begin{definition}[Unipotent]

An element \(u\) is \emph{unipotent} \(\iff\) \(u = 1+n\) where \(n\) is
nilpotent \(\iff\) its the only eigenvalue is \(\lambda = 1\).

\end{definition}

\begin{proposition}[JC Decomposition]

For any \(G\), there exists a closed embedding
\(G\hookrightarrow\operatorname{GL}(V) = \operatorname{GL}(n , k)\) and
for each \(x\in G\) a unique decomposition \(x=su\) where \(s\) is
semisimple (diagonalizable) and \(u\) is unipotent.

\end{proposition}

Define \(R_u(G)\) to be the subgroup of unipotent elements in \(R(G)\).

\begin{definition}[Semisimple and Reductive]

\hfill

Suppose \(G\) is connected, so \(G = G^0\), and nontrivial, so
\(G\neq \left\{{e}\right\}\). Then

\begin{itemize}
\tightlist
\item
  \(G\) is semisimple iff \(R(G) = \left\{{e}\right\}\).
\item
  \(G\) is reductive iff \(R_u(G) = \left\{{e}\right\}\).
\end{itemize}

\end{definition}

\begin{example}

\(G = \operatorname{GL}(n, k)\), then \(R(G) = Z(G) = kI\) the scalar
matrices, and \(R_u(G) = \left\{{e}\right\}\). So \(G\) is reductive and
semisimple.

\end{example}

\begin{example}

\(G = {\operatorname{SL}}(n , k)\), then \(R(G) = \left\{{I}\right\}\).

\begin{exercise}

Is this semisimple? Reductive? What is \(R_u(G)\)?

\end{exercise}

\end{example}

\begin{definition}[Torus]

A \emph{torus} \(T\subseteq G\) in \(G\) an algebraic group is a
commutative algebraic subgroup consisting of semisimple elements.

\end{definition}

\begin{example}

Let
\begin{align*}  
T \coloneqq
\left\langle{
\begin{bmatrix}
a_1 &  & \mathbf 0\\
 & \ddots &  \\
\mathbf 0 &  & a_n
\end{bmatrix} \subseteq \operatorname{GL}(n ,k)
}\right\rangle
.\end{align*}

\end{example}

\begin{remark}

Why are torii useful? For \(g = \mathrm{Lie}(G)\), we obtain a root
space decomposition
\begin{align*}  
g = 
\qty{\bigoplus_{\alpha \in \Phi_- }g_\alpha} \oplus 
t \oplus
\qty{\bigoplus_{\alpha \in \Phi_+ }g_\alpha} 
.\end{align*}

When \(G\) is a simple algebraic group, there is a
classification/correspondence:
\begin{align*}  
(G, T) \iff (\Phi, W)
.\end{align*}
where \(\Phi\) is an irreducible root system and \(W\) is a Weyl group.

\end{remark}

\hypertarget{monday-august-24}{%
\section{Monday, August 24}\label{monday-august-24}}

\hypertarget{review-and-general-setup}{%
\subsection{Review and General Setup}\label{review-and-general-setup}}

\begin{itemize}
\tightlist
\item
  \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
  is algebraically closed
\item
  \(G\) is a reductive algebraic group
\item
  \(T\subseteq G\) is a \emph{maximal split torus}
\end{itemize}

\begin{quote}
Split: \(T\cong \bigoplus {\mathbb{G}}_m\).
\end{quote}

We'll associate to this a root system, not necessarily irreducible,
yielding a correspondence
\begin{align*}  
(G, T) \iff (\Phi, W)
\end{align*}
with \(W\) a Weyl group.

This will be accomplished by looking at
\({\mathfrak{g}}= \mathrm{Lie}(G)\). If \(G\) is simple, then
\({\mathfrak{g}}\) is ``simple'', and \(\Phi\) irreducible will
correspond to a Dynkin diagram.

There is this a 1-to-1 correspondence
\begin{align*}  
G \text{ simple}/\sim \iff A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2
\end{align*}
where \(\sim\) denotes \emph{isogeny}.

Taking the Zariski tangent space at the identity ``linearizes'' an
algebraic group, yielding a Lie algebra.

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-24-14-09-33.png}
\caption{Image}
\end{figure}

We have the coordinate ring
\(k[G] = k[x_1, \cdots, x_n] / \mathcal{I}(G)\) where \(\mathcal{I}(G)\)
is the zero set. This is equal to \(\left\{{f:G\to k}\right\}\),

\hypertarget{the-associated-lie-algebra}{%
\subsection{The Associated Lie
Algebra}\label{the-associated-lie-algebra}}

\begin{definition}[The Lie Algebra of an Algebraic Group]

Define \emph{left translation} is
\begin{align*}  
\lambda_x: k[G] &\to k[G] \\
y &\mapsto f(x^{-1} y)
.\end{align*}

Define \emph{derivations} as
\begin{align*}  
\mathrm{Der} ~k[G] = \left\{{D: k[G] \to k[G] {~\mathrel{\Big|}~}D(fg) = D(f) g + f D(g) }\right\}
.\end{align*}

We can then realize the Lie algebra as
\begin{align*}  
{\mathfrak{g}}= \mathrm{Lie}(G) = \left\{{D\in \mathrm{Der} k[G] {~\mathrel{\Big|}~}\lambda_x \circ D = D\circ \lambda_x}\right\}
,\end{align*}
the left-invariant derivations.

\end{definition}

\begin{example}

\hfill

\begin{itemize}
\tightlist
\item
  \(G = \operatorname{GL}(n, k) \implies{\mathfrak{g}}= {\mathfrak{gl}}(n, k)\)
\item
  \(G = {\operatorname{SL}}(n, k) \implies{\mathfrak{g}}= {\mathfrak{sl}}(n, k)\)
\end{itemize}

\end{example}

Let \(G\) be reductive and \(T\) be a split torus. Then \(T\) acts on
\({\mathfrak{g}}\) via an \emph{adjoint action}. (For
\(\operatorname{GL}_n, {\operatorname{SL}}_n\), this is conjugation.)

There is a decomposition into eigenspaces for the action of \(T\),
\begin{align*}  
{\mathfrak{g}}= \qty{\bigoplus_{\alpha\in \Phi} g_\alpha} \oplus t
\end{align*}
where \(t = \mathrm{Lie}(T)\) and
\(g_\alpha \coloneqq\left\{{x\in {\mathfrak{g}}{~\mathrel{\Big|}~}t.x = \alpha(t) x\,\, \forall t\in T}\right\}\)
with \(\alpha: T\to K^{\times}\) a rational function (a \emph{root}).

In general, take \(\alpha\in\hom_{\text{AlgGrp}}(T, {\mathbb{G}}_m)\).

\begin{example}

Let \(G = \operatorname{GL}(n, k)\) and
\begin{align*}  
T = \left\{{
\begin{bmatrix}
a_1 & 0 & 0 \\
0 & \ddots & 0 \\
0 & 0 & a_n
\end{bmatrix}
{~\mathrel{\Big|}~}a_j\in k^{\times}
}\right\}
.\end{align*}

Then check the following action:

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-24-14-24-40.png}
\caption{Action}
\end{figure}

which indeed acts by a rational function.

Then
\begin{align*}  
g_\alpha = {\operatorname{span}}\left\{{
\begin{bmatrix}
0 & 1 & 0 \\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}
}\right\} = g_{(1, -1, 0)}
.\end{align*}

For \({\mathfrak{g}}= {\mathfrak{gl}}(3, k)\), we have
\begin{align*}  
{\mathfrak{g}}= t 
&\oplus g_{(1, -1, 0)}
\oplus g_{(-1, 1, 0)}  \\
&\oplus g_{(0, 1, -1)} 
\oplus g_{(0, -1, 1)}  \\
&\oplus g_{(1, 0, -1)} 
\oplus g_{(-1, 0, 1)} 
.\end{align*}

\end{example}

\hypertarget{representations}{%
\subsection{Representations}\label{representations}}

Let \(\rho: G\to \operatorname{GL}(V)\) be a group homomorphisms, then
equivalently \(V\) is a (rational) \(G{\hbox{-}}\)module.

For \(T\subseteq G\), \(T\curvearrowright G\) semisimply, so we can
simultaneously diagonalize these operators to obtain a \emph{weight
space decomposition} \(V = \bigoplus_{\lambda \in X(T)} V_\lambda\),
where
\begin{align*}  
V_\lambda &\coloneqq\left\{{v\in V{~\mathrel{\Big|}~}t.v = \lambda(t)v\,\, \forall t\in T}\right\} \\\
X(T) &\coloneqq\hom(T, {\mathbb{G}}_m)
.\end{align*}

\begin{example}

Let \(G = \operatorname{GL}(n, k)\) and \(V\) the
\(n{\hbox{-}}\)dimensional natural representation as column vectors,
\begin{align*}  
V = \left\{{{\left[ {v_1, \cdots, v_n} \right]} {~\mathrel{\Big|}~}v_j \in k}\right\}
.\end{align*}

Then
\begin{align*}  
T = \left\{{
\begin{bmatrix}
a_1 & 0 & 0 \\
0 & \ddots & 0\\
0 & 0 & a_n
\end{bmatrix} {~\mathrel{\Big|}~}a_j \in k^{\times}
}\right\}
.\end{align*}

Consider the basis vectors \(\mathbf{e}_j\), then
\begin{align*}  
\begin{bmatrix}
a_1 & 0 & 0 \\
0 & \ddots & 0\\
0 & 0 & a_n
\end{bmatrix} 
\begin{bmatrix}
0  \\
1  \\
0
\end{bmatrix}
=
a_j
\begin{bmatrix}
0  \\
1 \\
0
\end{bmatrix}
= a_1^0 a_2^0 \cdots a_j^0 \cdots a_n^0
\begin{bmatrix}
0  \\
1 \\
0
\end{bmatrix}
.\end{align*}

Here the weights are of the form
\({\varepsilon}_j\coloneqq{\left[ {0, 0, \cdots, 1, \cdots, 0} \right]}\)
with a \(1\) in the \(j\)th spot, so we have
\begin{align*}  
V = V_{{\varepsilon}_1} \oplus V_{{\varepsilon}_2} \oplus \cdots \oplus V_{{{\varepsilon}_n}}
.\end{align*}

\end{example}

\begin{example}

For \(V = {\mathbb{C}}\), we have \(t.v = (a_1^0 \cdots a_n^0)v\) and
\(V = V_{(0, 0, \cdots, 0)}\).

\end{example}

\hypertarget{classification}{%
\subsection{Classification}\label{classification}}

Let \(G\) be a simple algebraic group (ano closed, connected, normal
subgroups other than \(\left\{{e}\right\}, G\)) that is nonabelian that
is nonabelian.

\begin{example}

Let \(G = {\operatorname{SL}}(3, k)\). Then
\begin{align*}  
T = \left\{{
t = 
\begin{bmatrix}
a_1 & 0 & 0 \\
0 & a_1 a_2^{-1}  & 0\\
0 & 0 & a_2^{-1} 
\end{bmatrix}
{~\mathrel{\Big|}~}
a_1, a_2\in k^{\times}
}\right\}
\end{align*}
and
\begin{align*}  
t.
\begin{bmatrix}
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
= 
a_1^2 a_2^{-1} 
\begin{bmatrix}
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}
.\end{align*}
and \(\alpha_1 = (2, -1)\).

\todo[inline]{What is $\alpha_1$? Note that you can recover the Cartan something here?}

Then
\begin{align*}  
{\mathfrak{g}}= 
{\mathfrak{g}}_{(2, -1)} \oplus {\mathfrak{g}}_{(-2, 1)} \oplus
{\mathfrak{g}}_{(-1, 2)} \oplus {\mathfrak{g}}_{(1, -2)} \oplus
{\mathfrak{g}}_{(1, 1)} \oplus {\mathfrak{g}}_{(-1, -1)}
.\end{align*}

Then \(\alpha_2 = (-1, 2)\) and \(\alpha_1 + \alpha_2 = ( 1, 1)\).

This gives the root space decomposition for \({\mathfrak{sl}}_3\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-24-14-49-31.png}
\caption{Image}
\end{figure}

Then the Weyl group will be generated by reflections through these
hyperplanes.

\end{example}

\hypertarget{wednesday-august-26}{%
\section{Wednesday, August 26}\label{wednesday-august-26}}

\hypertarget{review}{%
\subsection{Review}\label{review}}

\begin{itemize}
\tightlist
\item
  \(G\) a reductive algebraic group over \(k\)
\item
  \(T = \prod_{i=1}^n {\mathbb{G}}_m\) a maximal split torus
\item
  \({\mathfrak{g}}= \mathrm{Lie}(G)\)
\item
  There's an induced root space decomposition
  \({\mathfrak{g}}= t\oplus \bigoplus_{\alpha\in \Phi}{\mathfrak{g}}_\alpha\)
\item
  When \(G\) is simple, \(\Phi\) is an \emph{irreducible} root system

  \begin{itemize}
  \tightlist
  \item
    There is a classification of these by Dynkin diagrams
  \end{itemize}
\end{itemize}

\begin{example}

\(A_n\) corresponds to \({\mathfrak{sl}}(n+1, k)\) (mnemonic: \(A_1\)
corresponds to \({\mathfrak{sl}}(2)\))

\end{example}

\begin{itemize}
\item
  We have representations \(\rho: G\to \operatorname{GL}(V)\),
  i.e.~\(V\) is a \(G{\hbox{-}}\)module
\item
  For \(T\subseteq G\), we have a weight space decomposition:
  \(V = \bigoplus_{\lambda \in X(T)} V_\lambda\) where
  \(X(T) = \hom(T, {\mathbb{G}}_m)\).

  \begin{quote}
  Note that \(X(T) \cong {\mathbb{Z}}^n\), the number of copies of
  \({\mathbb{G}}_m\) in \(T\).
  \end{quote}
\end{itemize}

\hypertarget{root-systems-and-weights}{%
\subsection{Root Systems and Weights}\label{root-systems-and-weights}}

\begin{example}

Let \(\Phi = A_2\), then we have the following root system:

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-26-14-05-58.png}
\caption{Image}
\end{figure}

\end{example}

In general, we'll have
\(\Delta = \left\{{\alpha_1, \cdots, \alpha_n}\right\}\) a basis of
\emph{simple roots}.

\begin{remark}

Every root \(\alpha\in I\) can be expressed as either positive integer
linear combination (or negative) of simple roots.

\end{remark}

For any \(\alpha\in \Phi\), let \(s_\alpha\) be the reflection across
\(H_\alpha\), the hyperplane orthogonal to \(\alpha\). Then define the
\emph{Weyl group}
\(W = \left\{{s_\alpha {~\mathrel{\Big|}~}\alpha\in \Phi}\right\}\).

\begin{example}

Here the Weyl group is \(S_3\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-26-14-10-24.png}
\caption{Image}
\end{figure}

\end{example}

\begin{remark}

\(W\) acts transitively on bases.

\end{remark}

\begin{remark}

\(X(T) \subseteq {\mathbb{Z}}\Phi\), recalling that
\(X(T) = \hom(T, {\mathbb{G}}_m) = {\mathbb{Z}}^n\) for some \(n\).
Denote \({\mathbb{Z}}\Phi\) the \emph{root lattice} and \(X(T)\) the
\emph{weight lattice}.

\end{remark}

\begin{example}

Let \(G = {\mathfrak{sl}}(2, {\mathbb{C}})\) then
\(X(T) = {\mathbb{Z}}\omega\) where \(\omega = 1\),
\({\mathbb{Z}}\Phi = {\mathbb{Z}}\left\{{\alpha}\right\}\) Then there is
one weight \(\alpha\), and the root lattice \({\mathbb{Z}}\Phi\) is just
\(2{\mathbb{Z}}\). However, the weight lattice is
\({\mathbb{Z}}\omega = {\mathbb{Z}}\), and these are not equal in
general.

\end{example}

\begin{remark}

There is partial ordering on \(X(T)\) given by
\(\lambda \geq \mu \iff \lambda - \mu = \sum_{\alpha\in \Delta} n_\alpha \alpha\)
where \(n_\alpha \geq 0\). (We say \(\lambda\) \emph{dominates}
\(\mu\).)

\end{remark}

\begin{definition}[Fundamental Dominant Weights]

We extend scalars for the weight lattice to obtain
\(E \coloneqq X(T) \otimes_{\mathbb{Z}}{\mathbb{R}}\cong {\mathbb{R}}^n\),
a Euclidean space with an inner product
\({\left\langle {{-}},~{{-}} \right\rangle}\).

For \(\alpha\in \Phi\), define its \emph{coroot}
\(\alpha {}^{ \vee }\coloneqq{2\alpha \over {\left\langle {\alpha},~{\alpha} \right\rangle}}\).
Define the \emph{simple coroots} as
\(\Delta {}^{ \vee }\coloneqq\left\{{\alpha_i {}^{ \vee }}\right\}_{i=1}^n\),
which has a dual basis
\(\Omega \coloneqq\left\{{\omega_i}\right\}_{i=1}^n\) the
\emph{fundamental weights}. These satisfy
\({\left\langle {\omega_i},~{\alpha_j {}^{ \vee }} \right\rangle} = \delta_{ij}\).

\todo[inline]{What is the notation for fundamental weights? Definitely not $\Omega$ usually!}

\begin{quote}
Important because we can index irreducible representations by
fundamental weights.
\end{quote}

A weight \(\lambda\in X(T)\) is \emph{dominant} iff
\(\lambda \in {\mathbb{Z}}^{\geq 0} \Omega\),
i.e.~\(\lambda = \sum n_i \omega_i\) with
\(n_i \in {\mathbb{Z}}^{\geq 0}\).

\end{definition}

If \(G\) is simply connected, then
\(X(T) = \bigoplus {\mathbb{Z}}\omega_i\).

\begin{quote}
See Jantzen for definition of simply-connected,
\({\operatorname{SL}}(n+1)\) is simply connected but its adjoint
\(PGL(n+1)\) is not simply connected.
\end{quote}

\hypertarget{complex-semisimple-lie-algebras}{%
\subsection{Complex Semisimple Lie
Algebras}\label{complex-semisimple-lie-algebras}}

When doing representation theory, we look at the Verma modules
\(Z(\lambda) = U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda \twoheadrightarrow L(\lambda)\).

\begin{theorem}[?]

\(L(\lambda)\) as a finite-dimensional
\(U({\mathfrak{g}}){\hbox{-}}\)module \(\iff\) \(\lambda\) is dominant,
i.e.~\(\lambda \in X(T)_+\).

\end{theorem}

Thus the representations are indexed by lattice points in a particular
region:

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-26-14-36-44.png}
\caption{Image}
\end{figure}

\textbf{Question 1}:

Suppose \(G\) is a simple (simply connected) algebraic group. How do you
parameterize \emph{irreducible} representations?

For \(\rho: G\to \operatorname{GL}(V)\), \(V\) is a \emph{simple module}
(an \emph{irreducible representation}) iff the only proper
\(G{\hbox{-}}\)submodules of \(V\) are trivial.

\textbf{Answer 1}: They are also parameterized by \(X(T)_+\). We'll show
this using the induction functor
\(\mathop{\mathrm{ind}}_B^G \lambda =H^0(G/B, \mathcal{L}(\lambda))\)
(sheaf cohomology of the flag variety with coefficients in some line
bundle).

\begin{quote}
We'll define what \(B\) is later, essentially upper-triangular matrices.
\end{quote}

\textbf{Question 2}: What are the dimensions of the irreducible
representations for \(G\)?

\textbf{Answer 2}: Over \(k={\mathbb{C}}\) using Weyl's dimension
formula.

For
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\):
conjectured to be known for \(p\geq h\) (the \emph{Coxeter number}), but
by Williamson (2013) there are counterexamples. Current work being done!

\hypertarget{friday-august-28}{%
\section{Friday, August 28}\label{friday-august-28}}

\hypertarget{representation-theory}{%
\subsection{Representation Theory}\label{representation-theory}}

Review: let \({\mathfrak{g}}\) be a semisimple lie algebra
\(/{\mathbb{C}}\). There is a decomposition
\({\mathfrak{g}}= {\mathfrak{b}}^+ \oplus {\mathfrak{n}}^- = {\mathfrak{n}}^+ \oplus t\oplus {\mathfrak{n}}^-\),
where \(t\) is a torus. We associate \(U({\mathfrak{g}})\) the universal
enveloping algebra, and representations of \({\mathfrak{g}}\) correspond
with representations of \(U({\mathfrak{g}})\).

Let \(\lambda \in X(T)\) be a weight, then \(\lambda\) is a
\(U({\mathfrak{b}}^+){\hbox{-}}\)module. We can write
\(Z(\lambda) = U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda\).

\begin{remark}

There exists a unique maximal submodule of \(Z(\lambda)\), say
\(RZ(\lambda)\) where \(Z(\lambda)/RZ(\lambda) \cong L(\lambda)\) is an
irreducible representation of \({\mathfrak{g}}\).

\end{remark}

\begin{theorem}[?]

Let \(L = L(\lambda)\) be a finite-dimensional irreducible
representation for \({\mathfrak{g}}\). Then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(L \cong Z(\lambda)/RZ(\lambda)\) for some \(\lambda\).
\item
  \(\lambda \in X(T)_+\) is a dominant integral weight.
\end{enumerate}

\end{theorem}

\hypertarget{induction}{%
\subsubsection{Induction}\label{induction}}

Let \({\mathfrak{g}}\) be an algebraic group \(/k\) with
\(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\),
and let \(H \leq G\). Let \(M\) be an \(H{\hbox{-}}\)module, we'll
eventually want to produce a \(G{\hbox{-}}\)modules.

Step 1: Make \(M\) into a \(G\times H\) where the first component
\((g, 1)\) acts trivially on \(M\).

Taking the coordinate algebra \(k[G]\), this is a
\((G-G){\hbox{-}}\)bimodule, and thus becomes a
\(G\times H{\hbox{-}}\)module. Let \(f\in k[G]\), so \(f:G\to K\), and
let \(y\in G\). The explicit action is
\begin{align*}  
[(g, h) f] (y) \coloneqq f(g^{-1} y h)
.\end{align*}

Note that we can identify \(H\cong 1\times H \leq G\times H\). We can
form \((M\otimes_k k[G])^H\), the \(H{\hbox{-}}\)fixed points.

\begin{exercise}

Let \(N\) be an \(A{\hbox{-}}\)module and \(B{~\trianglelefteq~}A\),
then \(N^B\) is an \(A/B{\hbox{-}}\)module.

\begin{quote}
Hint: the action of \(B\) is trivial on \(N^B\). Here
\(N^B \coloneqq\left\{{n\in N {~\mathrel{\Big|}~}b.n = n\, \forall b\in B}\right\}\)
\end{quote}

\end{exercise}

\begin{definition}[Induction]

The \emph{induced module} is defined as
\begin{align*}  
\mathop{\mathrm{ind}}_H^G(M) \coloneqq(M\otimes k[G])^H
.\end{align*}

\end{definition}

\hypertarget{properties-of-induction}{%
\subsubsection{Properties of Induction}\label{properties-of-induction}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(({-}\otimes_k k[G]) = \hom_H(k, {-}\otimes_k k[G])\) is only
  \emph{left-exact}, i.e.
  \begin{align*}  
  \qty{0\to A\to B\to C\to 0}\mapsto \qty{0\to FA \to FB \to FC \to \cdots}
  .\end{align*}
\item
  By taking right-derived functors \(R^jF\), you can take cohomology.
\end{enumerate}

\begin{quote}
Note that in this category, we won't have enough projectives, but we
will have enough injectives.
\end{quote}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{2}
\item
  This functor commutes with direct sums and direct limits.
\item
  (\textbf{Important}) Frobenius Reciprocity: there is an adjoint,
  \emph{restriction}, satisfying
  \begin{align*}  
  \hom_G(N, \mathop{\mathrm{ind}}_H^G M) = \hom_H(N\downarrow_H, M)
  .\end{align*}
\item
  (Tensor Identity) If \(M\in {\mathsf{Mod}}(H)\) and additionally
  \(M \in {\mathsf{Mod}}(G)\), then
  \(\mathop{\mathrm{ind}}_H^G = M \otimes_k \mathop{\mathrm{ind}}_H^G k\).
\end{enumerate}

If \(V_1, V_2 \in {\mathsf{Mod}}(G)\) then
\(V_1 \otimes_k V_2 \in {\mathsf{Mod}}(G)\) with the action given by
\(g(v_1\otimes v_2) = gv_1 \otimes gv_2\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{5}
\tightlist
\item
  Another interpretation: we can write
  \begin{align*}  
  \mathop{\mathrm{ind}}_H^G(M) = \left\{{f\in \mathop{\mathrm{Hom}}(G, M_a)
  {~\mathrel{\Big|}~}
  f(gh) = h^{-1} \cdot f(g)
  \, \forall g\in G, h\in H}\right\} \qquad M_a = M \coloneqq{\mathbb{A}}^{\dim M}
  .\end{align*}
\end{enumerate}

\begin{quote}
I.e., equivariant wrt the \(H{\hbox{-}}\)action.
\end{quote}

Then \(G\) acts on \(\mathop{\mathrm{ind}}_H^G M\) by left-translation:
\((gf)(y) = f(g^{-1} y)\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{6}
\tightlist
\item
  There is an evaluation map:
  \begin{align*}  
  {\varepsilon}: \mathop{\mathrm{ind}}_H^G(M) &\to M \\ 
  f&\mapsto f(1)
  .\end{align*}
\end{enumerate}

This is an \(H{\hbox{-}}\)module morphism. Why? We can check
\begin{align*}  
{\varepsilon}(h.f) 
&\coloneqq(h.f)(a) \\
&= f(h^{-1} ) \\
&= hf(1) \\
&= h({\varepsilon}(f))
.\end{align*}

We can write the isomorphism in Frobenius reciprocity explicitly:
\begin{align*}  
\hom_G(N, \mathop{\mathrm{ind}}_H^G M) &\xrightarrow{\cong} \hom_H(N, M) \\
\phi & \mapsto {\varepsilon}\circ \phi
.\end{align*}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{7}
\tightlist
\item
  Transitivity of induction: for \(H\leq H' \leq G\), there is a natural
  transformation (?) of functors:
  \begin{align*}  
  \mathop{\mathrm{ind}}_H^G({-}) = \mathop{\mathrm{ind}}_{H'}^G\qty{\mathop{\mathrm{ind}}_H^{H'}({-}) }
  .\end{align*}
\end{enumerate}

\todo[inline]{Equality as a composition of functors?}

\hypertarget{classification-of-simple-ghbox-modules}{%
\subsection{\texorpdfstring{Classification of Simple
\(G{\hbox{-}}\)modules}{Classification of Simple G\{\textbackslash hbox\{-\}\}modules}}\label{classification-of-simple-ghbox-modules}}

Suppose \(G\) is a connected reductive algebraic group \(/k\) with
\(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\).

\begin{example}

Let \(G = \operatorname{GL}(n, k)\). There is a decomposition:

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-28-14-39-50.png}
\caption{Image}
\end{figure}

\end{example}

\textbf{Step 1}: Getting modules for \(U\).

Then there's a general fact: \(U^+ T U \hookrightarrow G\) is dense in
the Zariski topology for any reductive algebraic group. We can form

\begin{itemize}
\tightlist
\item
  \(B^+ \coloneqq T\rtimes U^+\), the \emph{positive borel},
\item
  \(B^- \coloneqq T\rtimes U\), the \emph{negative borel},
\end{itemize}

Suppose we have a \(U{\hbox{-}}\)module, i.e.~a representation
\(\rho: U \to \operatorname{GL}(V)\). We can find a basis such that
\(\rho(u)\) is upper triangular with ones on the diagonal. In this case,
there is a composition series with 1-dimensional quotients, and the
composition factors are all isomorphic to \(k\).

Moral: for unipotent groups, there are only trivial representations,
i.e.~the only simple \(U{\hbox{-}}\)modules are isomorphic to \(k\).

\textbf{Step 2}: Getting modules for \(B\).

Modules for \(B\) are solvable, in which case we can find a flag. In
this case, \(\rho(b)\) embeds into upper triangular matrices, where the
diagonal action may now not be trivial (i.e.~diagonal is now arbitrary).

Thus simple \(B{\hbox{-}}\)modules arise by taking
\(\lambda \in X(T) = \hom(T, {\mathbb{G}}_m) = \hom(T, \operatorname{GL}(1, k))\),
then letting \(u\) act trivially on \(\lambda\), i.e.~\(u.v = v\). Here
we have \(B \to B/U = T\), so any \(T{\hbox{-}}\)module can be pulled
back to a \(B{\hbox{-}}\)module.

\textbf{Step 3}: Getting modules for \(G\).

Let \(\lambda \in X(T)\), then
\(H^0(\lambda) = \mathop{\mathrm{ind}}_B^G \lambda = \nabla(\lambda)\).

\hypertarget{monday-august-31}{%
\section{Monday, August 31}\label{monday-august-31}}

\hypertarget{review-of-representation-theory-of-modules}{%
\subsection{Review of Representation Theory of
Modules}\label{review-of-representation-theory-of-modules}}

Take \(R\) a ring, then consider \(M\) an \(R{\hbox{-}}\)module to be a
``vector space'' over \(M\). Note that \(M\) is an \(R{\hbox{-}}\)module
\(\iff\) there exists a ring morphism
\(\rho: R\to \hom_{\text{AbGrp}}(M, M)\).

Now let \(G\) be a group and consider \(G{\hbox{-}}\)modules \(M\). Then
a \(G{\hbox{-}}\)module will be defined by taking \(M/k\) a vector space
and a \(G{\hbox{-}}\)action on \(M\). This is equivalent to having a
group morphism \(\rho: G\to \operatorname{GL}(M)\).

For \(M\) a \(G{\hbox{-}}\)module, given a group action, define
\begin{align*}  
\rho: G&\to \operatorname{GL}(M) \\
\rho(g)(m) &= g.m
\end{align*}
where \(\rho(h): M\to M\).

Similarly, for \(\rho: G\to \operatorname{GL}(M)\) a group morphism,
define the group action \(g.m \coloneqq\rho(g)m\). Thus representations
of \(G\) and \(G{\hbox{-}}\)modules are equivalent.

\begin{definition}[?]

Let \(M\) be a \(G{\hbox{-}}\)module.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(M\) is a \emph{simple} \(G{\hbox{-}}\)module (equivalently an
  \emph{irreducible representation}) \(\iff\) the only
  \(G{\hbox{-}}\)submodules (equiv. \(G{\hbox{-}}\)invariant subspaces)
  are \(0, M\).
\item
  \(M\) is \emph{indecomposable} \(\iff\) \(M\) can not be written as
  \(M = M_1 \oplus M_2\) with \(M_i < M\) proper submodules.
\end{enumerate}

\end{definition}

\begin{example}

For \(G = {\operatorname{SL}}(n, {\mathbb{C}})\), there is a natural
\(n{\hbox{-}}\)dimensional representation \(M = V\), and this is
irreducible.

\end{example}

\todo[inline]{What is $V$?}

\begin{example}

Let \(R = {\mathbb{Z}}\), so we're considering
\({\mathbb{Z}}{\hbox{-}}\)modules. For \(M={\mathbb{Z}}\), \(M\) is not
simple since \(2{\mathbb{Z}}< {\mathbb{Z}}\) is a proper submodule.
However \(M\) is indecomposable.

\end{example}

Recall from last time: we defined a functor
\(\mathop{\mathrm{ind}}_H^G({-}): H{\hbox{-}}\text{mod} \to G{\hbox{-}}\text{mod}\),
where \(\mathop{\mathrm{ind}}_H^G = \qty{k[G] \otimes M}^H\), the
\(H{\hbox{-}}\)invariants. This functor is left-exact but not
right-exact, so we have cohomology \(R^j \mathop{\mathrm{ind}}_H^G\) by
taking right-derived functors.

Goal: classify simple \(G{\hbox{-}}\)modules for \(G\) a reductive
connected algebraic group.

\begin{example}

For \(G = \operatorname{GL}(n , k)\), we have a decomposition

\begin{figure}
\centering
\includegraphics{figures/image_2020-08-31-14-10-01.png}
\caption{Image}
\end{figure}

\end{example}

We have

\begin{itemize}
\tightlist
\item
  \(B = T\rtimes U\) the negative Borel,
\item
  \(B = T\rtimes U^+\) the Borel
\end{itemize}

For \(U{\hbox{-}}\)modules: \(k\) is the only simple
\(U{\hbox{-}}\)module. Importantly, if \(V\) is a \(U{\hbox{-}}\)module,
then the fixed points are never zero,
i.e.~\(V^U = \hom_{U{\hbox{-}}\text{Mod}}(k, V) \neq 0\).

For \(B{\hbox{-}}\)modules: let
\(X(T) \coloneqq\hom(T, {\mathbb{G}}_m) = \hom(T, \operatorname{GL}(1, k))\).
These are the simple representations for the torus \(T\). Thus
\(\lambda \in X(T)\) represents a simple \(T{\hbox{-}}\)module.

We have a map \(B \to B/U = T\), so we can pullback
\(T{\hbox{-}}\)representations to \(B{\hbox{-}}\)representations
(``inflation''), since we have a map \(T\to \operatorname{GL}(1, k)\)
and we can just compose. So \(\lambda\) is a 1-dimensional (simple)
\(B{\hbox{-}}\)module where \(U\) acts trivially.

Lee's theorem: all irreducible representations for \(B\) are
one-dimensional. Thus these are the simple \(B{\hbox{-}}\)modules.

For \(G{\hbox{-}}\)modules: define
\(\nabla(\lambda) \coloneqq\mathop{\mathrm{ind}}_B^G(\lambda) = H^0(\lambda)\).

Questions:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  When does \(H^0(\lambda) = 0\)?
\item
  What is \(\dim_{k{\hbox{-}}\text{Vect}} H^0(\lambda)\)?
\item
  What are the composition factors of \(H^0(\lambda)\)?
\end{enumerate}

\begin{quote}
Known in characteristic zero, wildly open in positive characteristic.
\end{quote}

\begin{remark}

Another interpretation: look at the flag variety \(G/B\) and take global
sections, then \(H^0(\lambda) = H^0(G/B, \mathcal{L}(\lambda))\) where
\(\mathcal{L}\) is given by projecting the fiber product
\(G \times_B \lambda \twoheadrightarrow G/B\) onto the first factor.

\end{remark}

\begin{remark}

\hfill

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(H^0(k) = H^0({\left[ {0, \cdots, 0} \right]}) = k[G/B] = k\).
\item
  \(H^0(M) = M\) if \(M\) is a \(G{\hbox{-}}\)module.
\item
  A \(G{\hbox{-}}\)module \(M\) is \emph{semisimple} iff
  \(M = \bigoplus_{i\in I} M_i\) with each \(M_i\) are simple.
\item
  Can consider the largest semisimple submodule, the \emph{socle}
  \(\mathop{\mathrm{Soc}}_G(M)\).
\end{enumerate}

\end{remark}

\begin{center}
\begin{tikzcd}
L_4 \ar[dr] & & L_5 \oplus L_7\ar[dl] \\
& \qty{L_1 \oplus L_2 \oplus L_3} = \mathop{\mathrm{Soc}}_G(M)) & \\
\end{tikzcd}
\end{center}

Goal: classify simple \(G{\hbox{-}}\)modules. Strategy: used dominant
highest weights.

\begin{quote}
As opposed to Verma modules, the irreducibles will be a dual situation
where they sit at the bottom of the module. Indicated by the notation
\(\nabla\) pointing down!
\end{quote}

\begin{proposition}[?]

Let \(\lambda \in X(T)\) with \(H^0(\lambda) \neq 0\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\dim H^0(\lambda)^{U^+} = 1\) and
  \(H^0(\lambda)^{U^+} = H^0(\lambda)_\lambda\).
\item
  Every weight of \(H^0(\lambda)\) satisfies
  \(w_u \lambda \leq \mu \leq \lambda\), where \(w_0\) is the longest
  Weyl group element and
  \(\alpha\leq \beta \iff \alpha-\beta \in {\mathbb{Z}}^{+}\Phi\).
\end{enumerate}

\begin{quote}
Note that in fact \(\ell(w_0) = {\left\lvert {\Phi^+} \right\rvert}\).
\end{quote}

\end{proposition}

\begin{example}

Take \(A_2\) with simple reflections \(s_{\alpha_1}, s_{\alpha_2}\) and
\(\Delta = \left\{{\alpha_1, \alpha_2}\right\}\).

\begin{center}
\begin{tikzcd}
& 1\ar[ld] \ar[rd] & \\
s_{\alpha_1} \ar[d] & & s_{\alpha_2}\ar[d] \\
s_{\alpha_1}s_{\alpha_2}\ar[rd] & & s_{\alpha_2} s_{\alpha_1}\ar[ld] \\
& s_{\alpha_2} s_{\alpha_1}s_{\alpha_1} = s_{\alpha_1} s_{\alpha_2} s_{\alpha_1} = w_0 & 
\end{tikzcd}
\end{center}

\end{example}

\begin{proof}[(Sketch)]

We can write
\begin{align*}  
H^0(\lambda) = \left\{{f\in k[G] {~\mathrel{\Big|}~}f(gb) = \lambda(b)^{-1} f(g) \, b\in B, g\in G}\right\}
.\end{align*}

Suppose \(f\in H^0(\lambda)^{U^+}\) and \(u_+ \in U^+, t\in T, u\in U\).
Then
\begin{align*}  
\qty{ u_+^{-1} f} (tu) 
&= f(tu) \\
&= \lambda(t)^{-1} f(1)
.\end{align*}
On the other hand,
\begin{align*}  
\qty{ u_+^{-1} f} (tu) 
&= f(u_+ t u)
.\end{align*}

So by density, \(f(1)\) is determined by \(f(u_+ t u)\) and
\(\dim H^0(\lambda)^{U^+} \leq 1\). But since this can't be zero, the
dimension must be equal to 1.

\end{proof}

\begin{proposition}[?]

Let
\begin{align*}  
{\varepsilon}: H^0(\lambda) \to \lambda
\end{align*}
be the evaluation morphism.

This is a morphism of \(B{\hbox{-}}\)modules, and in particular is a
morphism of \(T{\hbox{-}}\)modules. Thus the image of a weight
\(\mu \neq \lambda\) is zero, so \({\varepsilon}\) is injective.

\end{proposition}

\begin{proof}

We have
\begin{align*}  
f(u_+ t u) = \lambda(t)^{-1} f(1) = \lambda(t)^{-1} {\varepsilon}(f)
.\end{align*}

Suppose \(f\in H^0(\lambda)^{U^+}\) and \({\varepsilon}(f) = 0\). Then
\(f(u_+ t u) = 0\), and by density \(f\equiv 0\), showing injectivity.

Therefore \(H^0(\lambda)^{U^+}\subset H^0(\lambda)_\lambda\). Suppose
\(\mu\) is maximal among weights in \(H^0(\lambda)\). Then
\begin{align*}  
H^0(\lambda)_{\mu} \subseteq H^0(\lambda)^{U^+}
\end{align*}
because \(U^+\) raises weights.

But \(H^0(\lambda)^{U^+} \subseteq H^0(\lambda)_\lambda\) implies
\(\mu = \lambda\). Thus the maximal weight in \(H^0(\lambda)\) is
\(\lambda\).

\begin{quote}
Recall the situation in lie algebras:
\(g_\alpha v \in V_{\lambda + \alpha}\) when \(v\\in V_{\lambda}\).
\end{quote}

Since \(\lambda\) is maximal, any other weight \(\mu\) satisfies
\(\mu \leq \lambda\). Thus
\begin{align*}  
H^0(\lambda)_\lambda \subseteq H^0(\lambda)^{U^+} \subseteq H^0(\lambda)_\lambda
,\end{align*}
forcing these to be equal and finishing part 1.

\end{proof}

\hypertarget{friday-september-04}{%
\section{Friday, September 04}\label{friday-september-04}}

Some concepts used in the proof of other theorems: Let \(G\) be a
reductive algebraic group and \({\mathfrak{g}}\) its lie algebra. There
is an associative algebra \(U({\mathfrak{g}})\) which reflects the
representation theory of \(G\).

Fact: \({\mathfrak{g}}{\hbox{-}}\)mod
\(\equiv U({\mathfrak{g}}){\hbox{-}}\)modules which are unitary,
i.e.~\(1.m = m\).

We can write a basis
\begin{align*}  
{\mathfrak{g}}= \left\langle{e_\alpha, h_i, f_\beta {~\mathrel{\Big|}~}\alpha\in\Phi^+,\, \beta\in\Phi^-,\, i = 1,2,\cdots,n}\right\rangle
,\end{align*}
the \emph{Chevalley basis}. It turns out that the structure constants
are all in \({\mathbb{Z}}\).

\begin{example}

Take \({\mathfrak{g}}= {\mathfrak{sl}}(2, k)\), then
\begin{align*}  
e = \begin{bmatrix}
0 & 0 \\
1 & 0
\end{bmatrix}
\quad 
f = \begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}
\quad
h = \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
.\end{align*}

\end{example}

We want to form a \({\mathbb{Z}}{\hbox{-}}\)lattice in
\(U({\mathfrak{g}})\), denoted
\begin{align*}  
U({\mathfrak{g}})_{\mathbb{Z}}
=
\left\langle{
e_\alpha^{[n]} = {e_\alpha^n \over n!},\, f_\beta^{[n]} = {f_\beta^n \over n!}, {h_i \choose m}
}\right\rangle
.\end{align*}

We then form the \emph{distribution algebra} (or \emph{hyperalgebra} in
earlier literature) as
\(\mathrm{Dist})G) \coloneqq U({\mathfrak{g}})_{\mathbb{Z}}\otimes_{\mathbb{Z}}k\)
for \(k\) any field (e.g.~\(\operatorname{ch}(k) = p\)).

\begin{theorem}[?]

\(G{\hbox{-}}\)modules \(\equiv \mathrm{Dist}(G){\hbox{-}}\)modules
which are

\begin{itemize}
\tightlist
\item
  \emph{Weight modules}
\item
  \emph{Locally finite}: \(\dim \mathrm{Dist}(G).m < \infty\) for all
  \(m\in M\).
\end{itemize}

\end{theorem}

\begin{remark}

In characteristic zero, \(\mathrm{Dist}(G) = U({\mathfrak{g}})\). Thus
there is a correspondence
\begin{align*}  
\left\{{\substack{G{\hbox{-}}\text{modules}}}\right\} \iff
\left\{{\substack{U({\mathfrak{g}}){\hbox{-}}\text{modules}}}\right\} 
.\end{align*}

If \(\operatorname{ch}(k) = p\),
e.g.~\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\),
and if the Frobenius map \(F:G\to G\) satisfies \(G_1\coloneqq\ker F\)
(thinking of \(G_1\) as a group scheme), then
\(\mathrm{Dist}(G_1) < \mathrm{Dist}(G)\) is a proper submodule. In this
case, \({\mathfrak{g}}\subseteq \mathrm{Dist}(G_1)\) is a finite
dimensional Hopf algebra, and
\(k[G_1] = \mathrm{Dist}(G_1) {}^{ \vee }\). Importantly, the lie
algebra does \emph{not} generate \(\mathrm{Dist}(G)\) if
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\).

\end{remark}

\begin{example}

Take \(G = {\mathbb{G}}_a\), then
\(\mathrm{Dist}({\mathbb{G}}_a) = \left\langle{T^k {~\mathrel{\Big|}~}k=0,1,\cdots}\right\rangle\)
is an infinite dimensional algebra. In this case,
\(T^k T^\ell = {k+\ell \choose \ell}T^{k+\ell}\). For
\(k={\mathbb{C}}\),
\(\mathrm{Dist}({\mathbb{G}}_a) = \left\langle{T^1}\right\rangle\) has
one generator.

In the case
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\),
we have
\(\mathrm{Dist}(\qty{{\mathbb{G}}_a}_1) = \left\langle{T^k {~\mathrel{\Big|}~}0\leq k \leq p-1}\right\rangle\).

Note that taking duals yields a truncated polynomial algebra:
\(k[\qty{{\mathbb{G}}_a}_1] = k[x] / \left\langle{x^p}\right\rangle\).

\end{example}

\hypertarget{review-1}{%
\subsection{Review}\label{review-1}}

Recall that \(H^0(\lambda) \coloneqq\mathop{\mathrm{ind}}_B^G \lambda\).
Proved in last (missed) class:

\begin{remark}

Let \(H^0(\lambda) \neq 0\). Then

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(\dim H^0(\lambda)_\lambda = 1\) where
  \(H^0(\lambda) = H^0(\lambda)^{U^+}\).
\item
  Each weight \(\mu\) of \(H^0(\lambda)\) satisfies
  \(w_0 \lambda \leq \mu \leq \lambda\), where \(w_0\) is the longest
  Weyl group element.
\end{enumerate}

\end{remark}

\begin{remark}

Let \(H^0(\lambda)_\lambda \neq 0\), then
\(L(\lambda) = \mathop{\mathrm{Soc}}_G H^0(\lambda)\) is simple.

\end{remark}

\begin{remark}

If \(\mu\) is a weight of \(L(\lambda)\), then
\(w_0 \lambda \leq \mu \leq \lambda\), \(\dim L(\lambda)_\lambda = 1\),
and \(L(\lambda)_\lambda = L(\lambda)^{U+}\).

\end{remark}

\begin{remark}

Any simple \(G{\hbox{-}}\)module is isomorphic to \(L(\lambda)\) where
\(H^0(\lambda) \neq 0\).

\end{remark}

Goal: We now want to classify simple \(G{\hbox{-}}\)modules. So we need
an iff criterion for when \(H^0(\lambda) \neq 0\).

We look at the set of dominant weights
\begin{align*}  
X(T)_+ 
&= \left\{{\lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}\geq 0 \forall \alpha\in\Delta}\right\}
&= \left\{{\lambda \in X(T) {~\mathrel{\Big|}~}\lambda = \sum_{i=1}^\ell n_i w_i,\, n_i \geq 0}\right\}
.\end{align*}

\begin{theorem}[?]

TFAE:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(H^0(\lambda) \neq 0\).
\item
  \(\lambda \in X(T)_+\), i.e.~\(\lambda\) is a dominant weight.
\end{enumerate}

\end{theorem}

\begin{proof}

\(1\implies 2\): Suppose (1), then consider a simple reflection
\(s_\alpha\) for some \(\alpha \in \Delta\). We know
\(H^0(\lambda)_\lambda \neq 0\), thus
\(H^0(\lambda)_{s_\alpha \lambda} \neq 0\). Therefore
\begin{align*}  
s_\alpha \lambda = \lambda - {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}\alpha \leq \lambda \\
\implies 0 \leq {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}\alpha \\
\implies {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \geq 0 \qquad \forall \alpha\in \Delta
.\end{align*}

\(2\implies 1\): For a detailed proof, see Jantzen 2.6 in Part II.

\begin{itemize}
\item
  Let \(\lambda \in X(T)_+\), then (by the intro lie algebras course)
  there exists an \(L(\lambda)\): a simple finite dimensional
  \(U({\mathfrak{g}}){\hbox{-}}\)module over \({\mathbb{C}}\).
\item
  \(L(\lambda)\) has an integral basis which is compatible with
  \(U({\mathfrak{g}})_{\mathbb{Z}}\) (Kostant's
  \({\mathbb{Z}}{\hbox{-}}\)form).
\item
  Thus we can base change to get
  \(\tilde L(\lambda) \coloneqq L(\lambda) \otimes_{\mathbb{Z}}k\),
  which is a \(\mathrm{Dist}(G){\hbox{-}}\)module. Note that
  \(\tilde L(\lambda)\) still has highest weight \(\lambda\), so
  consider \(\hom_B(\tilde L(\lambda), \lambda) \neq 0\).
\item
  Apply frobenius reciprocity:
  \(\hom_B(\tilde L(\lambda), \lambda) = \hom_G(\tilde L(\lambda), \mathop{\mathrm{ind}}_B^G \lambda) = \hom_G(\tilde L(\lambda), H^0(\lambda))\).
  But then \(H^0(\lambda) \neq 0\) (since otherwise this would imply the
  original hom was zero).
\end{itemize}

\end{proof}

\begin{theorem}[?]

Let \(G\) be a reductive connected algebraic group over \(k\). Then
there exists a 1-to-1 correspondence between dominant weights and
irreducible \(G{\hbox{-}}\)representations:
\begin{align*}  
\left\{{\substack{\text{Dominant weights: } X(T)_+}}\right\} \iff
\left\{{\substack{\text{Irreducible representations: }\left\{{L(\lambda) {~\mathrel{\Big|}~}\lambda \in X(T)_+}\right\} }}\right\}
.\end{align*}

\end{theorem}

\hypertarget{characters-of-ghbox-modules}{%
\subsection{\texorpdfstring{Characters of
\(G{\hbox{-}}\)modules}{Characters of G\{\textbackslash hbox\{-\}\}modules}}\label{characters-of-ghbox-modules}}

Let \(G\) be reductive, so (importantly) it has a maximal torus \(T\).
Let \(M\in G{\hbox{-}}\mathrm{mod}\), so (importantly)
\(M\in T{\hbox{-}}\mathrm{mod}\).

Then there is a weight space decomposition
\(M = \bigoplus_{\lambda \in X(T)} M_\lambda\). We then write the
character of \(M\) as
\begin{align*}  
\operatorname{ch}M \coloneqq\sum_{\lambda \in X(T)} \qty{\dim M_\lambda} e^{\lambda} \in {\mathbb{Z}}[X(T)]
.\end{align*}

Next time: more characters, and Weyl's dimension formula.

\hypertarget{wednesday-september-09}{%
\section{Wednesday, September 09}\label{wednesday-september-09}}

Todo

\hypertarget{wednesday-september-16}{%
\section{Wednesday, September 16}\label{wednesday-september-16}}

\hypertarget{group-schemes}{%
\subsection{Group Schemes}\label{group-schemes}}

\begin{definition}[Representable Functors]

Let \(F:: {\mathsf{Alg}_{/k} }\to {\mathsf{Set}}\) be a functor, then
\(F\) is \textbf{representable} iff \(F(R)\) corresponds to ``solutions
to equations in \(R\)''.

\end{definition}

\begin{example}

Let \(F({-}) = {\operatorname{SL}}(2, {-})\), then the corresponding
equations are \(\operatorname{det}(x_{ij}) = 1\).

\end{example}

If \(F\) is representable, there is a correspondence
\(F(R) \cong \hom_R(A, R)\). In the above example,
\begin{align*}A = k[x_{11}, x_{12}, x_{21}, x_{22}] / \left\langle{x_{11} x_{22} - x_{12}x_{21}}\right\rangle,\end{align*}
which is exactly the coordinate algebra.

\begin{definition}[Affine Group Scheme]

An \emph{affine group scheme} is a representable functor
\(F:{\mathsf{Alg}_{/k} }\to{\mathsf{Group}}\).

\end{definition}

Suppose \(G\) is an affine group scheme, and let \(A = k[G]\) be the
representing object. Then there is a correspondence
\begin{align*}  
G{\hbox{-}}\text{modules} \iff k[G] {}^{ \vee }{\hbox{-}}\text{modules}
.\end{align*}

For \(G\) reductive, the RHS is equivalent to
\(\operatorname{Dist}(G){\hbox{-}}\)modules.

\begin{definition}[Finite Group Schemes]

\(G\) is a \textbf{finite} group scheme iff \(k[G]\) is finite
dimensional.

\end{definition}

If \(G\) is finite, then \(A {}^{ \vee }\cong k[G] {}^{ \vee }\) is a
cocommutative Hopf algebras. Thus representations for \emph{finite}
group schemes are equivalent to representations for finite-dimensional
cocommutative Hopf algebras.

\begin{quote}
On group scheme side: see reduction, spectral sequences, conceptual
arguments. On the algebra side: have bases, underlying vector space, can
do concrete computations. Can take
\(\operatorname{Spec}\qty{k[G]} {}^{ \vee }\) to recover a group scheme.
\end{quote}

\hypertarget{hopf-algebras}{%
\subsection{Hopf Algebras}\label{hopf-algebras}}

For \(A\) a \({\mathsf{Alg}_{/k} }\), we have a multiplication and a
unit, which can be defined in terms of diagrams. To categorically
reverse arrows, we can ask for a comultiplication and a counit.
\begin{align*}  
\Delta: A &\to A^{\otimes 2}
\\ \\
\epsilon: A &\to k 
.\end{align*}

We'll want another map, an \emph{antipode}
\begin{align*}  
s: A\to A
.\end{align*}

The comultiplication should satisfy

\begin{center}
\begin{tikzcd}
A^{\otimes 3} & \ar[l, "1\otimes A"] A^{\otimes 2} \\
A^{\otimes 2}\ar[u, "\Delta \otimes 1"] & \ar[l, "\Delta"]\ar[u, "\Delta"] A
\end{tikzcd}
\end{center}

The counit should satisfy

\begin{center}
\begin{tikzcd}
k\otimes\ar[d, "\cong"] A & \ar[l, "{{\varepsilon}\otimes 1}"] A^{\otimes 2}\\
A\ar[r, "\cong"] & A\ar[u, "\Delta"]
\end{tikzcd}
\end{center}

And the antipode should satisfy

\begin{center}
\begin{tikzcd}
A & A\ar[l, "{m(s\otimes 1)}"] \\
A\ar[u] & A\ar[l, "{\varepsilon}"] \ar[u, "\Delta"]
\end{tikzcd}
\end{center}

\hypertarget{module-constructions}{%
\subsubsection{Module Constructions}\label{module-constructions}}

Let \(A\) be a Hopf algebra.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  For \(A{\hbox{-}}\)modules \(M, N\), we can form the
  \(A{\hbox{-}}\)module \(M\otimes_k N\) with
  \begin{align*}  
  \Delta(a) &= \sum a_i \otimes a_j \\ \\
  a(m\otimes n) &= \sum a_1 m \otimes a_2 n
  .\end{align*}
\item
  If \(M\) is finite-dimensional over \(A\), then
  \(M {}^{ \vee }= \hom_k(M, k) \ni f\) is an \(A{\hbox{-}}\)module, and
  we can define \((af)(x) \coloneqq f(s(a)x)\) for \(a\in A, x\in M\).
\end{enumerate}

\begin{example}

\(A = kG\) the group algebra on a group is a Hopf algebra:
\begin{align*}  
\Delta: A &\to A^{\otimes 2} \\
g &\mapsto g\otimes g
.\end{align*}

The module action is diagonal, namely \(g(m\otimes n) = gm \otimes gn\).
The antipode is given by \(s(g) = g^{-1}\), and the unit is
\({\varepsilon}(g) = 1\) for all \(g\in G\).

\end{example}

\begin{example}

Let \(A = U({\mathfrak{g}})\), the universal enveloping algebra for
\({\mathfrak{g}}\) a Lie algebra. Recall that
\({\mathfrak{g}}{\hbox{-}}\)modules are equivalent to
\(U({\mathfrak{g}}){\hbox{-}}\)modules (unitary representations, some
big associative algebra). Then \(A\) is a Hopf algebra, with
\(\Delta(\ell) = \ell\otimes 1 + 1\otimes\ell\) for
\(\ell \in {\mathfrak{g}}\). The unit is \({\varepsilon}(\ell) = 0\),
and the antipode is \(s(\ell) = -\ell\).

\end{example}

\begin{example}

Take the additive group \({\mathbb{G}}_a\), then
\(A = k[{\mathbb{G}}_a] \cong k[x]\) is a commutative Hopf algebra with
\(\Delta(x) = x\otimes 1 + 1\otimes x\),
\({\varepsilon}(x) = 0, s(x) = -x\).

\end{example}

\begin{example}

For \({\mathbb{G}}_m\), we have
\(A = k[{\mathbb{G}}_m] \cong k[x, x^{-1}], {\varepsilon}(x) = 1, s(x) = x^{-1}\).

\end{example}

\hypertarget{frobenius-kernels}{%
\subsection{Frobenius Kernels}\label{frobenius-kernels}}

Let \(G\) be an algebraic group (scheme) over \(k\), where
\(\operatorname{ch}(k) = p\). Let \(F:G\to G\) be the Frobenius, where
e.g.
\begin{align*}  
F:\operatorname{GL}(n, {-}) &\to \operatorname{GL}(n, {-})\\
(x_{ij}) & \mapsto (x_{ij}^p)
.\end{align*}

Then \(F\) is a map of group schemes.

\begin{definition}[Frobenius Kernels]

\(G_r \coloneqq\ker F^r\), where
\(F^r \coloneqq F\circ F \circ \cdots \circ F\) is the
\(r{\hbox{-}}\)fold composition of the Frobenius.

This yields a nesting
\(G_1 {~\trianglelefteq~}G_2 {~\trianglelefteq~}G_3 \cdots \leq G\).

\end{definition}

Recall that
\begin{align*}  
\operatorname{Dist}(G) = \left\langle{ {x_\alpha^n \over n!}, {y_\beta^m \over m!}, {H_i \choose k} }\right\rangle
.\end{align*}

We get a chain of finite dimensional algebras
\begin{align*}  
\operatorname{Dist}(G_1) \leq \operatorname{Dist}(G_2) \leq \cdots \leq \operatorname{Dist}(G)
\end{align*}
where
\begin{align*}  
\operatorname{Dist}(G_1) = \left\langle{ {x_\alpha^n \over n!}, {y_\beta^m \over m!}, {H_i \choose k} {~\mathrel{\Big|}~}0\leq n,m,k \leq p-1 }\right\rangle
,\end{align*}

where in general \(\operatorname{Dist}(G_\ell)\) goes up to
\(p^{\ell} - 1\). Recall that \(G_r\) representations were equivalent to
\(\operatorname{Dist}(G_r)\) representations.

Some basic questions (Curtis, Steinberg, 1960s):

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  What are the simple modules for Frobenius kernels? I.e., what are the
  irreducible representations for \(G_r\)?
\item
  How are the representations for \(G_r\) related to those for \(G\)?
\end{enumerate}

\begin{quote}
It turns out the representations for \(G_r\) will lift to
representations to \(G\). Use ``twisted tensor product'' (Steinberg).
\end{quote}

\begin{remark}

It turns out that \(G_1\) is special.
\begin{align*}  
\operatorname{Dist}(G_1) \cong u({\mathfrak{g}}) \coloneqq U({\mathfrak{g}}) / \left\langle{x^p - x^{[p]}}\right\rangle
,\end{align*}
where \({\mathfrak{g}}= \mathrm{Lie}(G)\) is a \emph{restricted lie
algebra} (N. Jacobson). Note that for \(D\in {\mathfrak{g}}\) a
derivation, we define \(D^{[p]} \coloneqq D\circ \cdots \circ D\) is the
\(p{\hbox{-}}\)fold composition.

\(G_1{\hbox{-}}\)modules are equivalent to
\({\mathfrak{g}}{\hbox{-}}\)modules which are \emph{restricted} in the
sense that
\begin{align*}  
\rho: g &\to {\mathfrak{gl}}(V) \\
x^{[p]} &\mapsto \rho(x)^p
.\end{align*}

\end{remark}

\hypertarget{friday-september-18}{%
\section{Friday, September 18}\label{friday-september-18}}

\hypertarget{frobenius-kernels-1}{%
\subsection{Frobenius Kernels}\label{frobenius-kernels-1}}

Let \(\operatorname{ch}(k) p > 0\) and let \(G\) be an algebraic group
scheme. We have a Frobenius map \(F:G\to G\) given by
\(F((x_{ij})) = (x_{ij}^p)\), which we can iterate to get \(F^r\) for
\(r\in {\mathbb{N}}\). Setting \(G_r = \ker F^r\) the \(r\)th Frobenius
kernel, we get a normal series of group schemes
\begin{align*}  
G_1 {~\trianglelefteq~}G_2 {~\trianglelefteq~}\cdots {~\trianglelefteq~}G
.\end{align*}

There is an associated chain of finite dimensional Hopf algebras
\begin{align*}  
\operatorname{Dist}(G_1) \leq \operatorname{Dist}(G_2) \leq \cdots \leq \operatorname{Dist}(G)
.\end{align*}

Then \(k[G] {}^{ \vee }= \operatorname{Dist}(G_r)\), and we get an
equivalence of representations for \(G_r\) to representations for
\(\operatorname{Dist}(G_r)\).

A special case will be when \(G\) is a reductive algebraic group scheme.
We'll start by finding a basis for \(\operatorname{Dist}(G_r)\).

Recall the PBW theorem: we have a basis for \({\mathfrak{g}}\) given by
\begin{align*}  
\left\{{x_\alpha {~\mathrel{\Big|}~}\alpha\in \Phi^+ }\right\} &\text{ Positive root vectors} \\
\left\{{h_i {~\mathrel{\Big|}~}i=1,\cdots, n}\right\} &\text{ A basis for } t \\
\left\{{x_\alpha {~\mathrel{\Big|}~}\alpha\in \Phi^- }\right\} &\text{ Negative root vectors} \\
.\end{align*}

We can then obtain a basis for \(U({\mathfrak{g}})\):
\begin{align*}  
U({\mathfrak{g}}) = \left\langle{ \prod_{\alpha\in\Phi^+} x_\alpha^{n(\alpha)} \prod_{i=1}^n h_i^{k_i} \prod_{\alpha\in\Phi^+} x_{-\alpha}^{m(\alpha)}  }\right\rangle
.\end{align*}

We can similarly obtain a basis for the distribution algebra
\begin{align*}  
\operatorname{Dist}(G) = \left\langle{ 
\prod_{\alpha\in\Phi^+} { x_{\alpha}^{n(\alpha)} \over n!} 
\prod_{i=1}^n {h_i \choose k_i} 
\prod_{\alpha\in\Phi^+} { x_{-\alpha}^{n(\alpha)} \over n!} 
}\right\rangle
,\end{align*}
and we can similar get \(\operatorname{Dist}(G_r)\) by restricting to
\(0\leq n(\alpha), k_i, m(\alpha) \leq p^r - 1\). Above the \(k_i\) are
allowed to be any integers. This yields a triangular decomposition
\begin{align*}  
\operatorname{Dist}(G_r) = \operatorname{Dist}(U_r^+) \operatorname{Dist}(T_r) \operatorname{Dist}(U_r^-)
,\end{align*}
where we'll denote the first two terms \(\operatorname{Dist}(B_r^+)\)
and the last two as \(\operatorname{Dist}(B_r)\).

\hypertarget{induced-and-coinduced-modules}{%
\subsection{Induced and Coinduced
Modules}\label{induced-and-coinduced-modules}}

Goal: Classify simple \(G_r{\hbox{-}}\)modules. We know the
classification of simple \(G{\hbox{-}}\)modules, so we'll follow similar
reasoning. We started by realizing
\(L(\lambda) \hookrightarrow\mathop{\mathrm{ind}}_B^G \lambda\) as a
submodule (the socle) of some ``universal'' module.

Let \(M\) be a \(B_r{\hbox{-}}\)module, we can then define
\begin{align*}  
\mathop{\mathrm{ind}}_{B_r}^{G_r}M = \qty{k[G_r] \otimes M }^{B_r}
,\end{align*}
where we're now taking the \(B_r{\hbox{-}}\)invariants. We get a
decomposition as vector spaces,
\begin{align*}  
k[G_r] = k[U_r^+] \otimes_k k[B_r]
\end{align*}
and thus an isomorphism
\begin{align*}  
\mathop{\mathrm{ind}}_{B_r}^{G_r}M = \qty{k[G_r] \otimes M }^{B_r} 
\cong k[U_r^+] \otimes\qty{ k[B_r] \otimes M}^{B_r}
\cong k[U_r^+] \otimes M
\end{align*}
since
\(k[B_r]\otimes M \cong \mathop{\mathrm{ind}}_{B_r}^{B_r} M \cong M\).

We then define
\begin{align*}  
\mathop{\mathrm{coInd}}_{B_r}^{G_r} = \operatorname{Dist}(G_r) \otimes_{\operatorname{Dist}(B_r)} \otimes M
,\end{align*}
which is an analog of
\(U({\mathfrak{g}})\otimes_{U({\mathfrak{b}})} M\).

We have
\(\operatorname{Dist}(U_r^+) \otimes\operatorname{Dist}(B_r) \cong \operatorname{Dist}(G_r)\),
so

\begin{align*}  
\mathop{\mathrm{coInd}}_{B_r}^{G_r} = \operatorname{Dist}(G_r) \otimes_{\operatorname{Dist}(B_r)} \otimes M
\cong
\operatorname{Dist}(U_r^+) \otimes_k \operatorname{Dist}(B_r) \otimes_{\operatorname{Dist}(B_r)} M 
\cong
\operatorname{Dist}(U_r^+) \otimes_k M
,\end{align*}
which we'll define as the \textbf{coinduced module}.

We can compute the dimension:
\begin{align*}  
\dim \mathop{\mathrm{ind}}_{B_r}^{G_r} M = \dim \mathop{\mathrm{coInd}}_{B_r}^{G_r} M = \qty{\dim M} p^{r{\left\lvert {\Phi^+} \right\rvert}}
.\end{align*}

\begin{quote}
Open: don't know how to compute composition factors.
\end{quote}

\begin{proposition}[?]

\hfill

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item

  \begin{align*}\mathop{\mathrm{coInd}}_{B_r}^{G_r} M \equiv \mathop{\mathrm{ind}}_{B_r}^{G_r} M\otimes 2(p^r - 1)\rho,\end{align*}
  where the last term is a one-dimensional \(B_r{\hbox{-}}\)module and
  \(\rho\) is the \emph{Weyl weight}.
\item

  \begin{align*}\mathop{\mathrm{coInd}}_{B_r^+}^{G_r} M \cong \mathop{\mathrm{ind}}_{B_r^+}^{G_r} M \otimes-2\qty{p^r-1}\rho\end{align*}
\end{enumerate}

where
\begin{align*}  
\rho = {1\over 2}\sum_{\alpha\in\Phi^+} \alpha = \sum_{i=1}^n w_i
.\end{align*}

\end{proposition}

\begin{proof}[Sketch for (1)]

Since the tensor product satisfies a universal property, we have a map

\begin{center}
\begin{tikzcd}
M \ar[rd, "B_r"]\ar[rr] & & \operatorname{Dist}(G_r)\otimes_{\operatorname{Dist}(B_r)} M\ar[dl, "\exists \psi", dotted] \\
& N = M\mathop{\mathrm{ind}}_{B_r}^{G_r} \otimes 2(p^r-1)\rho &
\end{tikzcd}
\end{center}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  We need to find a \(B_r\) morphism \(f:M\to N\).
\item
  We need to show that \(f\) generates \(N\) as a
  \(G_r{\hbox{-}}\)module.
\end{enumerate}

Note that if (1) and (2) hold, then \(\psi\) is surjective, but since
\(\dim \mathop{\mathrm{coInd}}_{B_r}^{G_r} M= \dim N\) this forces
\(\psi\) to be an isomorphism.

We can write
\begin{align*}  
\mathop{\mathrm{ind}}_{B_r}^{G_r} M\otimes 2(p^r-1) \rho
&=
\qty{ k[G_r] \otimes M \otimes 2(p^r-1) \rho  }^{B_r} \\
&\cong
\hom_{B_r}\qty{\operatorname{Dist}(G_r), M\otimes 2(p^r-1)\rho }
.\end{align*}

Let \(g_m(x) \coloneqq m\otimes 2(p^r-1)\rho\) for any
\(x =\prod_{\alpha\in\Phi^+} {x_\alpha^{p^r-1} \over \qty{p^r-1}! }\),
and \(g_m(x) = 0\) for any other \(x\).

Now define \(f(m) = g_m\), and check that \(\operatorname{im}f\)
generates \(N\).

\end{proof}

\hypertarget{verma-modules}{%
\subsection{Verma Modules}\label{verma-modules}}

Recall that
\(W(\lambda) \coloneqq U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda\)
were the \emph{Verma modules} for lie algebras.

Let \(\lambda \in X(T)\), we have \(T_r \leq T\) and restriction yields
a map \(X(T) \to X(T_r)\). Given a weight \(\lambda\), we can write it
\(p{\hbox{-}}\)adically as
\begin{align*}  
\lambda = \lambda_0 + \lambda_1 p + \lambda_2 p^2 + \cdots + \lambda_{r-1} + \cdots
.\end{align*}

This yields an exact sequence
\begin{align*}  
0 \to p^r X(T) \to X(T) \to X(T_r) \to 0
,\end{align*}

and thus \(X(T) / p^r X(T) \cong X(T_r)\).

Let \(\lambda \in X(T_r)\), then \(\lambda\) becomes a
\(B_r{\hbox{-}}\)module by letting \(U_r\) act trivially, since we have
\begin{align*}  
\cdots U_r \to B_r \twoheadrightarrow T_r \to 0
.\end{align*}

Set \(Z(r) = \mathop{\mathrm{coInd}}_{B_r}^{G_r} \lambda\), and set
\(Z(r)' = \mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda\). Then
\(\dim Z_r(\lambda) = \dim Z_r'(\lambda) = p^{r{\left\lvert {\Phi^+} \right\rvert}}\).
We'll then think of

\begin{itemize}
\tightlist
\item
  \(\mathop{\mathrm{coInd}}\twoheadrightarrow L_r(\lambda)\) being in
  the head,
\item
  \(L_r(\lambda) \hookrightarrow\mathop{\mathrm{ind}}\) being the socle.
\end{itemize}

\begin{quote}
Note that the dimensions aren't known, nor are the projective covers or
injective hulls.
\end{quote}

We have a form of translation invariance, namely
\begin{align*}  
Z_r(\lambda + p^r\nu) = Z_r(\lambda) \qquad &\forall \nu \in X(T) \\
Z_r'(\lambda + p^r\nu) = Z_r'(\lambda) \qquad &\forall \nu \in X(T)
.\end{align*}

\begin{proposition}[?]

Let \(\lambda \in X(T)\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(Z_r(\lambda){\downarrow}_{B_r}\) is the projective cover of
  \(\lambda\) and the injective hull of \(\lambda - 2(p^r-1)\rho\).
\item
  \(Z_r'(\lambda){\downarrow}_{B_r^+}\) is the injective hull of
  \(\lambda\) and the projective hull of \(\lambda - 2(p^r-1)\rho\).
\end{enumerate}

\end{proposition}

\hypertarget{monday-september-21}{%
\section{Monday, September 21}\label{monday-september-21}}

Let \(G\) be a reductive algebraic group scheme,
\(k=\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\)
with \(p>0\), equipped with the Frobenius map \(F:G\to G\) with \(F^r\)
its \(r{\hbox{-}}\)fold composition. We defined \emph{Frobenius kernels}
\(G_r \coloneqq\ker F^r\), which are in correspondence with the
cocommutative Hopf algebras \(\operatorname{Dist}(G_r)\).

Goal: We want to classify simple \(G_r{\hbox{-}}\)modules, and to do
this we'll use socles.

We have a maximal torus \(T\subseteq G\) and thus \(T_r \subseteq G_r\)
after acting by Frobenius. This yields a SES
\begin{align*}  
0 \to p_r X(T) \to X(T) \to X(T)/p^r X(T) = X(T_r) \to 0
.\end{align*}

How to think about this: take \(\lambda \in X(T_r)\), then we can write
\(\lambda = \lambda + p^r \sigma\) in \(X(T_r)\) for some other weight
\(\sigma \in X(T)\). We'll define the ``baby Verma modules''
\begin{align*}  
Z_r(\lambda) \coloneqq\mathop{\mathrm{coInd}}_{B_r^+}^{G_r} \lambda \\
Z_r'(\lambda) \coloneqq\mathop{\mathrm{ind}}_{B_r^+}^{G_r} \lambda
,\end{align*}

and we have
\(\dim Z_r(\lambda) = \dim Z_r'(\lambda) = p^{r {\left\lvert {\Phi^+} \right\rvert}}\).

\begin{proposition}[?]

Let \(\lambda\in X(T)\) be a weight.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(Z_r(\lambda)\downarrow_{B_r}\) is the \emph{projective cover} of
  \(\lambda\) and the \emph{injective hull} of
  \(\lambda - 2 (p^r-1) \rho\).
\item
  \(Z_r'(\lambda)\downarrow_{B_r^+}\) is the \emph{injective hull} of
  \(\lambda\) and the \emph{projective cover} of
  \(\lambda - 2 (p^r-1) \rho\).
\end{enumerate}

\end{proposition}

\begin{quote}
Note the latter are \(T_r{\hbox{-}}\)modules, so we let \(U^+\) act
trivially.
\end{quote}

\begin{proof}[of 1]

What we need to do:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show \(Z_r(\lambda)\downarrow_{B_r}\) is projective.
\item
  Show \(Z_r(\lambda)\) is the smallest projective module such that
  \(Z_r(\lambda) \twoheadrightarrow\lambda\).
\end{enumerate}

For (1), we can write
\begin{align*}
\operatorname{Dist}(G_r) = \operatorname{Dist}(U_r^+) \operatorname{Dist}(B_r) = \operatorname{Dist}(B_r^+) \operatorname{Dist}(U_r),
,\end{align*}
and so
\begin{align*}  
Z_r(\lambda) 
&= \mathop{\mathrm{coInd}}_{B_r^+}^{G_r} \lambda \\
&= \qty{\operatorname{dist}(G_r) \otimes_{\operatorname{Dist}(B_r)} \lambda} \downarrow_{B_r^+} \\
&= \operatorname{Dist}(U_r^+)\otimes\lambda \\
&= \operatorname{Dist}(B_r^+) \otimes_{\operatorname{Dist}(T_r)} \lambda \\
&= \mathop{\mathrm{coInd}}_{T_r}^{B_r^+} \lambda
.\end{align*}

Why is this projective? Look at cohomology, suffices to show that higher
Exts vanish. So consider
\begin{align*}  
\operatorname{Ext} _{B_r^+}^n(\mathop{\mathrm{coInd}}_{T_r}^{B_r^+}, M) 
&= \operatorname{Ext} _{T_r}^n (\lambda, M) \qquad\text{by Frobenius reciprocity} \\
&= 0 \qquad \text{for } n \geq 0
,\end{align*}
since representations for \(T_r\) are completely reducible, and we've
used the fact that \(\mathop{\mathrm{coInd}}_{T_r}^{B_r^+}({-})\) is
exact.

\begin{quote}
Note: general algebra fact that higher exts vanish for projective
modules.
\end{quote}

For (2), we can write
\begin{align*}  
\hom_{B_r^+}(Z_r(\lambda), \mu)
&= \hom_{B_r^+}(\mathop{\mathrm{coInd}}_{T_r}^{B_r^+} \lambda, \mu) \\
&= \hom_{T_r} (\lambda, \mu) \qquad\text{by Frobenius reciprocity} \\
&=
\begin{cases}
k \& \lambda = \mu \\
0 \& \text{else}.
\end{cases}
\end{align*}

Thus
\(Z_r(\lambda) / \mathop{\mathrm{Rad}}Z_r(\lambda) \downarrow{B_r^+} = \lambda\).

If we now write \(A= \operatorname{Dist}(B_r^+)\) and
\({\mathfrak{g}}= {\mathfrak{n}}^+ \oplus t \oplus {\mathfrak{n}}\) with
\({\mathfrak{b}}^+ \coloneqq{\mathfrak{n}}^+ \oplus t\),
\begin{align*}
\sum_S \qty{\dim P(S)} \qty{\dim(S)} \\
&= \sum_{\lambda \in X(T_r)} \qty{\dim Z_r(\lambda)} \qty{\dim \lambda} \\
&= \sum_{\lambda \in X(T_r)} p^{r{\left\lvert {\Phi^+} \right\rvert}} \cdot 1 \\
&= {\left\lvert {X(T_r)} \right\rvert} p^{r{\left\lvert {\Phi^+} \right\rvert}} \\
&= p^{rn} p^{r{\left\lvert {\Phi^+} \right\rvert}} \qquad n = \dim t\\
&= p^{r \dim {\mathfrak{b}}^+} \\
&= \dim A
\end{align*}

\end{proof}

\hypertarget{simple-ghbox-modules}{%
\subsection{\texorpdfstring{Simple
\(G{\hbox{-}}\)modules}{Simple G\{\textbackslash hbox\{-\}\}modules}}\label{simple-ghbox-modules}}

We know that after taking fixed points, \(Z_r(\lambda)^{U_r}\) and
\(Z_r'(\lambda)^{U_r^+}\) are one-dimensional, and thus
\begin{align*}  
Z_r(\lambda) / \mathop{\mathrm{Rad}}Z_r(\lambda) \cong L_r(\lambda) \qquad \mathop{\mathrm{Soc}}_{G_r} Z_r'(\lambda) = L_r(\lambda)
\end{align*}
following the same argument considering \(H_0(\lambda)\).

For any \(\lambda \in X(T_r)\) we have
\(0\neq L_r = \mathop{\mathrm{Soc}}_{G_r} Z_r'(\lambda)\). By the
one-dimensionality above, we know
\begin{align*}  
L_r(\mu) = L_r(\lambda) \iff \lambda = \mu \in X(T_r)
.\end{align*}

Letting \(N\) be a simple \(G_r{\hbox{-}}\)module, we can consider it as
a \(B_r{\hbox{-}}\)module, and the simple \(B_r{\hbox{-}}\)modules are
one dimensional and obtained from simple \(T_r{\hbox{-}}\)modules. We
then know that for some \(\lambda \in X(T_r)\),
\begin{align*}  
0 \neq \hom_{B_r}(N, \lambda) \\
&= \hom_{G_r}(N, \mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda)
,\end{align*}
which implies that
\(N\hookrightarrow\mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda = Z_r'(\lambda)\)
as a submodule, and thus \(N = L_r(\lambda)\).

\begin{theorem}[Main Theorem]

Let \(\Lambda\) be a set of representatives of
\(XX(T) / p^r X(T) \cong X(T_r)\). Then there exists a one-to-one
correspondence
\begin{align*}  
\Lambda \iff \left\{{L_r(\lambda) \lambda \in \Lambda}\right\}
,\end{align*}
where the RHS are simple \(G_r{\hbox{-}}\)modules.

\end{theorem}

How to think about this: \textbf{restricted regions}. Choose dominant
weights as representatives
\begin{align*}  
X_r(T) 
&= \left\{{\lambda \in X(T)_+ {~\mathrel{\Big|}~}0\leq {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} < p^r\, \forall \alpha\in \Delta }\right\} \\
&= \left\{{\lambda \in X(T)_+ {~\mathrel{\Big|}~}\lambda = \sum_{i=1}^\ell n_i w_i,\, 0\leq n_j \leq p^r-1\, \forall j}\right\} \\
.\end{align*}

Pictures:

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-21-14-43-23.png}
\caption{Root systems, chambers formed by dominant weights}
\end{figure}

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-21-14-43-56.png}
\caption{Restricted regions}
\end{figure}

Some facts:

If \(\lambda \in X(T)_+\), then \(L(\lambda)\) is a simple
\(G{\hbox{-}}\)module.

\textbf{Question 1}: What happens when we restrict
\(L(\lambda)\downarrow_{G_r}\)?

\textbf{Answer}: This remains irreducible over \(G_r\) iff
\(\lambda \in X_r(T)\), i.e.~if
\(L(\lambda)\downarrow_{G} \cong L_r(\lambda)\) when
\(\lambda \in X_r(T)\).

\textbf{Question 2}: Given \(L(\lambda)\) for \(\lambda \in X(T)_+\),
can we express \(L(\lambda)\) in terms of simple
\(G_r{\hbox{-}}\)modules?

\textbf{Answer}: Yes, can be formulated in terms of \emph{Steinberg's
twisted tensor product}.

\hypertarget{friday-september-25}{%
\section{Friday, September 25}\label{friday-september-25}}

\hypertarget{review-and-proposition}{%
\subsection{Review and Proposition}\label{review-and-proposition}}

From last time: Steinberg's tensor product.

Let \(G\) be a reductive algebraic group scheme over \(k\) with
\(\operatorname{ch}(k) > 0\). We have a Frobenius \(F:G\to G\), we
iterate to obtain \(F^r\) and examine the Frobenius kernels
\(G_r\coloneqq\ker F^r\).

If we have a representation \(\rho: G\to \operatorname{GL}(M)\), we can
``twist'' by \(F^r\) to obtain
\(\rho^{(r)}: G \to \operatorname{GL}(M^{(r)})\). We have

Here \(M^{(r)}\) has the same underlying vector space as \(M\), but a
new module structure coming from \(\rho^{(r)}\). Note that \(G_r\) acts
trivially on \(M^{(r)}\).

\begin{itemize}
\tightlist
\item
  \(\left\{{L(\lambda) {~\mathrel{\Big|}~}\lambda \in X(T)_+}\right\}\)
  are the simple \(G{\hbox{-}}\)modules,
\item
  \(\left\{{L_r(\lambda) {~\mathrel{\Big|}~}\lambda \in X_r(T)_+}\right\}\)
  are the simple \(G_r{\hbox{-}}\)modules,
\end{itemize}

Note that \(L(\lambda)\downarrow_{G_r}\) is semisimple, equal to
\(L_r(\lambda)\) for \(\lambda \in X_r(T)\).

\begin{quote}
1960's, Curtis and Steinberg.
\end{quote}

\begin{proposition}[?]

Let \(\lambda \in X_r(T)\) and \(\mu \in X(T)_+\). Then
\begin{align*}  
L(\lambda + p^r \mu) \cong L(\lambda) \otimes L(\mu)^{(r)}
.\end{align*}

\end{proposition}

Recall that socle formula: letting \(M\) be a \(G{\hbox{-}}\)module, we
have an isomorphism of \(G{\hbox{-}}\)modules:
\begin{align*} 
\mathop{\mathrm{Soc}}_{G_r} \cong \bigoplus_{\lambda \in X_r(T)} L(\lambda) \otimes\hom_{G_r}(L(\lambda), M)
.\end{align*}

\hypertarget{proof}{%
\subsection{Proof}\label{proof}}

\begin{proof}

Let \(M = L(\lambda + p^r \mu)\). Then from the socle formula, only one
summand is nonzero, and thus \(\hom_{G_r}(L(\lambda), M)\) must be
simple. Then there exists a \(\tilde \lambda \in X_r(T)\) and a
\(\tilde \mu \in X(T)_+\) such that
\begin{align*}  
M = L(\tilde \lambda) \otimes L(\tilde\mu)^{(r)}
.\end{align*}

We now compare highest weights:
\begin{align*}  
\lambda + p^r \mu = \tilde \lambda + p^r \tilde \mu \implies \lambda = \tilde \lambda {\quad \operatorname{and} \quad} \mu = \tilde \mu
.\end{align*}

\end{proof}

\begin{theorem}[Steinberg]

Let \(\lambda \in X(T)_+\), with a \(p{\hbox{-}}\)adic expansion
\begin{align*}  
\lambda = \lambda_0 + \lambda_1 p + \cdots + \lambda_m p^m
.\end{align*}
where \(\lambda_j \in X_1(T)\) for all \(j\). Then
\begin{align*}  
L(\lambda) = L(\lambda_0) \otimes\bigotimes_{j=1}^m L(\lambda_j)^{(j)}
.\end{align*}

\end{theorem}

\begin{corollary}[?]

In order to know \(\dim L(\lambda)\) for \(\lambda \in X(T)_+\), it is
enough to know \(\dim L_1(\mu)\) for \(\mu \in X_1(T)\). Schematic:

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-25-14-13-47.png}
\caption{Image}
\end{figure}

\end{corollary}

\hypertarget{some-history}{%
\subsection{Some History}\label{some-history}}

Recall that simplie \(G_1{\hbox{-}}\)modules correspond to simple
\(\operatorname{Dist}(G_1){\hbox{-}}\)modules, and
\(\operatorname{Dist}(G_1) \cong U({\mathfrak{g}})\).

\begin{itemize}
\item
  1980: Lusztig proved conjecture: \(\operatorname{ch}L(\lambda)\) for
  \(\lambda \in X_1(T)\) is given by KL polynomials, shown for
  \(p \geq 2(h-1)\).
\item
  Kato showed for \(p> h\), where \(h\) is the \emph{Coxeter number}
  satisfying
  \(h = {\left\langle {\rho},~{\alpha_i {}^{ \vee }} \right\rangle} + 1\)
  where \(\alpha_i {}^{ \vee }\) is the highest short root.
\item
  1990's: A relation to representations of quantum groups \(U_q\) and
  affine lie algebras \(\widehat{{\mathfrak{g}}}\):

  \begin{center}
  \begin{tikzcd}
  \pmod u({\mathfrak{g}}) & \ar[l] \pmod U_q({\mathfrak{g}}) \ar[r, "\cong"] & \pmod\widehat{{\mathfrak{g}}}
  \end{tikzcd}
  \end{center}

  The first map is due to Andersen-Jantzen-Soergel for \(p\gg 0\) with
  no effective lower bounds, and the equivalence is due to
  Kazhdan-Lusztig, where the L conjecture holds for
  \(\widehat{{\mathfrak{g}}}\).
\item
  2000's: Fiebig showed the L conjecture holds for \(p>N\) where \(N\)
  is an effective (but large) lower bound.
\item
  2013: Geordie Williamson shows L conjecture is false, with infinitely
  many counterexamples, and no lower bounds that are linear in \(h\).
\end{itemize}

\begin{quote}
See Donkin's Tilting Module conjecture: expected that characters may
come from \(p{\hbox{-}}\)KL polynomials instead.
\end{quote}

\begin{example}

Let \(G= {\operatorname{SL}}(2)\), so \(\dim T =1\). Here the restricted
region of weights is given by
\(X_!(T) = \left\{{0,1,\cdots, p-1}\right\}\). Then
\(H^0(\lambda) = S^\lambda(V)\) for
\(\lambda \in X(T)_+ = {\mathbb{Z}}_{\geq 0}\) and
\(L(\lambda) \subseteq H^0(\lambda)\).

\begin{theorem}[?]

\begin{align*}  
L(\lambda) =  H^0(\lambda) {\quad \operatorname{for} \quad} \lambda \in X_1(T)
.\end{align*}

\end{theorem}

\begin{theorem}[?]

\begin{align*}  
\dim L(\lambda) = \lambda + 1 {\quad \operatorname{for} \quad} \lambda \in X_1(T)
.\end{align*}

\end{theorem}

Take \(p=3\). Then \(\dim L(0) = 1\), \(\dim L(1) = 2\) (the natural
representation), and \(\dim L(2) = 3\) (the adjoint representation).
Then for \(p=4\), we have to use the twisted tensor product formula.
Taking the 3-adic expansion \(4 = 1\cdot 3^0 + 1\cdot 3^1\), we have
\begin{align*}  
L(4) = L(1) \otimes L(1)^{(1)}
.\end{align*}

Since \(\dim L(1) = 2\), we get \(\dim L(4) = 4\).

Similarly, considering \(7 = 1\cdot 3^0 + 2\cdot 3^1\), we get
\begin{align*}  
L(7) \cong L(1) \otimes L(2)^{(1)}
\end{align*}
and so \(\dim L(7) = 6\).

Take \(p=5\), then

\begin{itemize}
\tightlist
\item
  \(\dim L(0) = 1\)
\item
  \(\dim L(1) = 2\)
\item
  \(\dim L(2) = 3\)
\item
  \(\dim L(3) = 4\)
\item
  \(\dim L(4) = 5\)
\end{itemize}

What is \(H^0(5)\)? We know \(L(5)\) is a submodule, and we can write
the character
\begin{align*}  
\operatorname{ch}H^0(5) = e^5 + e^3 + e^1 + e^{-1} + e^{-3} + e^{-5}
.\end{align*}

We know \(\operatorname{ch}(L(1)) = e^1 + e^{-1}\) and
\(L(5) = L(1)^{(1)}\), so we can write
\(\operatorname{ch}L() = e^{5} + e^{-5}\). By quotienting, we have
\(\operatorname{ch}H^0(5) - \operatorname{ch}L(5) = e^3 + e^1 + e^{-1} +e^{-3} = \operatorname{ch}L(3)\).
Thus the composition factors of \(H^0(5)\) are \(L(5)\) and \(L(3)\).

These correspond to an action of the affine Weyl group:

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-25-14-45-28.png}
\caption{Image}
\end{figure}

There is a \textbf{strong linkage principle} which describes the
possible composition factors of \(H^0(\lambda)\).

We can thus find the socle/head structure:

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-25-14-47-44.png}
\caption{Image}
\end{figure}

Thus \(\operatorname{Ext} _G^1(L(5), L(3)) \cong k\).

\end{example}

\begin{quote}
Note that in other types, we don't know the characters of the
irreducibles in the restricted region, so we don't necessarily know the
composition factors.
\end{quote}

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-25-14-51-40.png}
\caption{Image}
\end{figure}

\hypertarget{monday-september-28}{%
\section{Monday, September 28}\label{monday-september-28}}

\hypertarget{kempfs-theorem}{%
\subsection{Kempf's Theorem}\label{kempfs-theorem}}

Next topic: Kempf's Vanishing Theorem. Proof in Jantzen's book involving
ampleness for sheaves.

Setup:

We have

\begin{center}
\begin{tikzcd}
G & \text{a reductive algebraic group over } k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu \\
B\ar[u, "\subseteq"] & \text{the Borel subgroup} \\
T\ar[u, "\subseteq"] & \text{its maximal torus}
\end{tikzcd}
\end{center}

along with the weights \(X(T)\).

We can consider derived functors of induction, yielding
\(R^n \mathop{\mathrm{ind}}_B^G \lambda = \mathcal{H}^n(G/B, \mathcal{L}(\lambda)) \coloneqq H^n(\lambda)\)
where \(\mathcal{L}(\lambda)\) is a line bundle and \(G/B\) is the flag
variety.

Recall that

\begin{itemize}
\tightlist
\item
  \(H^0(\lambda) = \mathop{\mathrm{ind}}_B^G(\lambda)\),
\item
  \(\lambda \not\in X(T)_+ \implies H^0(\lambda) = 0\)
\item
  \(\lambda \in X(T)_+ \implies L(\lambda) = \mathop{\mathrm{Soc}}_G H^0(\lambda) \neq 0\).
\end{itemize}

\begin{theorem}[Kempf]

If \(\lambda \in X(T)_+\) a dominant weight, then \(H^n(\lambda) = 0\)
for \(n> 0\).

\end{theorem}

\begin{remark}

In \(\operatorname{ch}(k) = 0\), \(H^n(\lambda)\) is known by the
Bott-Borel-Weil theorem. In positive characteristic, this is not know:
the characters \(\operatorname{ch}H^n (\lambda)\) is known, and it's not
even known if or when they vanish. Wide open problem!

\begin{quote}
Could be a nice answer when \(p>h\) the Coxeter number.
\end{quote}

\end{remark}

\hypertarget{good-filtrations-and-weyl-filtrations}{%
\subsection{Good Filtrations and Weyl
Filtrations}\label{good-filtrations-and-weyl-filtrations}}

We define two classes of distinguished modules for
\(\lambda \in X(T)_+\):

\begin{itemize}
\tightlist
\item
  \(\nabla(\lambda) \coloneqq H^0(\lambda) = \mathop{\mathrm{ind}}_B^G \lambda\)
  the costandard/induced modules.
\item
  \(\Delta(\lambda) = V(\lambda) \coloneqq H^0(-w_0 \lambda) = \mathop{\mathrm{ind}}_B^G \lambda\)
  the standard/Weyl modules

  \begin{itemize}
  \tightlist
  \item
    Here \(w_0\) is the longest element in the Weyl group
  \end{itemize}
\end{itemize}

We have
\begin{align*}  
L(\lambda) &\hookrightarrow\nabla(\lambda)
\Delta(\lambda) &\twoheadrightarrow L(\lambda)
.\end{align*}

We define the category \(\text{Rat}{\hbox{-}}G\) of rational
\(G{\hbox{-}}\)modules. This is a \emph{highest weight category} (as is
e.g.~Category \({\mathcal{O}}\)).

\begin{definition}[Good Filtrations]

An (possibly infinite) ascending chain of \(G{\hbox{-}}\)modules
\begin{align*}  
0 \leq V_0 \subseteq V_1 \subseteq V_2 \subseteq \cdots \subseteq V
\end{align*}
is a \textbf{good filtration} of \(V\) iff

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(V = \cup_{i\geq 0} V_i\)
\item
  \(V_i/V_{i-1} \cong H^0(\lambda_i)\) for some
  \(\lambda_i \in X(T)_+\).
\end{enumerate}

\begin{quote}
In characteristic zero, the \(H^0\) are irreducible and this recovers a
composition series. Since we don't have semisimplicity in this category,
this is the next best thing.
\end{quote}

\end{definition}

\begin{definition}[Weyl Filtration]

With the same conditions of a good filtration, a chain is a \textbf{Weyl
filtration} on \(V\) iff

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(V = \cup_{i\geq 0} V_i\)
\item
  \(V_i/V_{i-1} \cong V(\lambda_i)\) for some \(\lambda_i \in X(T)_+\).
\end{enumerate}

\begin{quote}
I.e. the different is now that the quotients are standard modules.
\end{quote}

\end{definition}

\begin{definition}[Tilting Modules]

\(V\) is a \textbf{tilting module} iff \(V\) has both a good filtration
and a Weyl filtration.

\end{definition}

\begin{theorem}[Ringel, 1990s]

Let \(\lambda \in X(T)_+\) be a dominant weight. Then there is a unique
indecomposable highest weight tilting module \(T(\lambda)\) with highest
weight \(\lambda\).

\end{theorem}

\begin{example}

We have the following situation for type \(A_2\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-28-14-18-03.png}
\caption{Image}
\end{figure}

And thus a decomposition:

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-28-14-18-46.png}
\caption{Image}
\end{figure}

\end{example}

The picture to keep in mind is the following: 4 types of modules, all
indexed by dominant weights:

\begin{center}
\begin{tikzcd}
& H^0(\lambda) & \\
L(\lambda) \ar[ur, hookrightarrow] & & T(\lambda)\arrow[ul, twoheadrightarrow]\\
& V(\lambda) \arrow[ul, twoheadrightarrow] \ar[ur, hookrightarrow]
\end{tikzcd}
\end{center}

\hypertarget{cohomological-criteria-for-good-filtrations}{%
\subsection{Cohomological Criteria for Good
Filtrations}\label{cohomological-criteria-for-good-filtrations}}

We'll take cohomology in the following way: let \(G\) be an algebraic
group scheme, and define
\begin{align*}  
H^n(G, M) \coloneqq
\mathrm{Ext} G^n(k, M)
\end{align*}

where to compute \(\operatorname{Ext} _G^n(M, N)\) we take an injective
resolution \(N \hookrightarrow I_*\), apply \(\hom_G(M, {-})\), and take
kernels mod images.

Letting \(\lambda \in {\mathbb{Z}}\Phi\) be integral, so
\(\lambda_{\alpha \in \Delta} = \sum n_\alpha \alpha\), define the
\textbf{height}
\begin{align*}  
\operatorname{ht}(\lambda) = \sum_{\alpha\in\Delta} n_\alpha
.\end{align*}

\begin{lemma}[?]

There exists an injective resolution of \(B{\hbox{-}}\)modules
\begin{align*}  
0\to k\to I_0 \to I_1 \to \cdots
\end{align*}
where

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(I_0\) is the injective hull of \(k\),
\item
  All weights of \(I_j\), say \(\mu\) satisfy
  \(\operatorname{ht}(\mu) \geq j\).
\end{enumerate}

\end{lemma}

\begin{align*}  
k[u] \text{ an injective $B{\hbox{-}}$module} \\
k\hookrightarrow\mathop{\mathrm{ind}}_T^B k \coloneqq I_0 = k[u]
.\end{align*}

We thus get a diagram of the form

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-28-14-32-38.png}
\caption{Image}
\end{figure}

\begin{proposition}[?]

Let \(H\leq G\), then there exists a spectral sequence
\begin{align*}  
E^{i, j}_2 = \operatorname{Ext} _G^i(N, R^j \mathop{\mathrm{ind}}_H^G M) \implies \operatorname{Ext} _H^{i+j}(N, M)
\end{align*}
for \(N\in {\mathsf{Mod}}(G), M\in {\mathsf{Mod}}(H)\).

\end{proposition}

\begin{example}

Let \(H=B\) and take \(G=G\) itself, and let \(N = k\) the trivial
module and \(M\in {\mathsf{Mod}}(G)\) be any rational
\(G{\hbox{-}}\)module. We have
\begin{align*}  
E_2^{i, j} = \operatorname{Ext} ^{i}_B(k, R^j \mathop{\mathrm{ind}}_B^G M) \implies \operatorname{Ext} ^{i+j}_B(k, M)
.\end{align*}

Observations:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{-1}
\item
  \(R^0 \mathop{\mathrm{ind}}_B^G k = \mathop{\mathrm{ind}}_B^G k = k\).
\item
  The tensor identity works here,
  i.e.~\(R^j \mathop{\mathrm{ind}}_B^G M = \qty{R^j \mathop{\mathrm{ind}}_B^G k} \otimes M\).
\item
  \(R^j \mathop{\mathrm{ind}}_B^G k = 0\) for \(j> 0\) since we have a
  dominant weight.
\end{enumerate}

The spectral sequence thus collapses on \(E_2\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-28-14-41-33.png}
\caption{Image}
\end{figure}

Thus
\begin{align*}  
E_2^{i, 0} = \operatorname{Ext} ^i_B(k, M) = H^i(B, M)
.\end{align*}

\end{example}

\begin{corollary}[?]

Let \(G \supseteq P \supseteq B\) where \(P\) is a \emph{parabolic}
subalgebra and let \(M\) be a rational \(G{\hbox{-}}\)module. Then
\(H^n(G, M) = H^n(P, M) = H^n(B, M)\) for all \(n \geq 0\).

\end{corollary}

\begin{example}

Fix a Dynkin diagram and take a subset \(J\subseteq \Delta\).

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-28-14-47-01.png}
\caption{i}
\end{figure}

Then \(L_j\rtimes U_j = P_J = P\), and we have a decomposition like

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-28-15-13-36.png}
\caption{Image}
\end{figure}

\end{example}

\begin{proposition}[?]

Let \(M\in {\mathsf{Mod}}(P)\) with \(P\supseteq B\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\dim M < \infty\) then \(\dim H^n(P, M) < \infty\) for all \(n\).
\item
  If \(H^j(P, M) \neq 0\) then there exists \(\lambda\) a weight of
  \(M\) with \(-\lambda \in {\mathbb{N}}\Phi^+\) and
  \(\operatorname{ht}(-\lambda) \geq j\).
\end{enumerate}

\end{proposition}

\hypertarget{wednesday-september-30}{%
\section{Wednesday, September 30}\label{wednesday-september-30}}

Recall that we had a dominant weight \(\lambda \in X(T)_+\) with

\begin{center}
\begin{tikzcd}
& V(\lambda)\ar[dl, "\twoheadrightarrow"]\ar[dr, "\hookrightarrow"] & \\
L(\lambda)\ar[dr, "\hookrightarrow"] & &T(\lambda)\ar[dl, "\twoheadrightarrow"] \\
& H^0(\lambda) &
\end{tikzcd}
\end{center}

where we have a module with both a \emph{good} and a \emph{Weyl}
filtration.

If \(B\subseteq P \subseteq G\) with \(P\) parabolic and
\(M\in {\mathsf{Mod}}(G)\), we have a ``transfer theorem'': maps
\begin{align*}  
H^n(G; M) \xrightarrow{\mathop{\mathrm{res}}} H^n(P; M) \xrightarrow{\mathop{\mathrm{res}}} H^n(B; M)
\end{align*}
induced by restrictions which are isomorphisms.

\begin{proposition}[?]

Let \(M\in {\mathsf{Mod}}(P)\) with \(P\supseteq B\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\dim M < \infty\) then \(\dim H^n(P; M) < \infty\).
\item
  If \(H^j(P; M) \neq 0\) then there exists a weight \(\lambda\) of
  \(M\) such that \(-\lambda \in {\mathbb{N}}\Phi^+\) and
  \(\operatorname{ht}(-\lambda) \geq j\).
\end{enumerate}

\end{proposition}

\begin{quote}
Part (a) is proved in the book, we won't show it here.
\end{quote}

\begin{proof}[of part b]

Suppose \(H^j(P; M) \neq 0\), then we have an injective resolution
\(I_*\) for \(k\). Tensoring with \(M\) yields an injective resolution
for \(M\),
\begin{align*}  
0 \to M \to I_0\otimes M \to I_1 \otimes M \to \cdots
.\end{align*}
Since \(H^j(B; M) \neq 0\), we know that the cocycles
\(\hom_B(k, I_j\otimes M) \neq 0\) and thus
\(\hom_T(k, I_j\otimes M) \neq 0\).

So there exists a weight \(-\lambda\) of \(I_j\) with
\(\operatorname{ht}(-\lambda) \geq j\), and we know \(\lambda\) is a
weight of \(M\) applying the previous lemma: namely we know that
\(\lambda\) is invariant under the torus action, so there is a weight
\(-\lambda\) such that \(-\lambda + \lambda = 0\).

\end{proof}

\todo[inline]{? Why the last part?}

\begin{theorem}[?]

Let \(\lambda, \mu \in X(T)_+\), then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  The cohomology in the tensor product is zero, except in one special
  case:
  \begin{align*}  
  H^i(G, H^0(\lambda) \otimes H^0(\mu))
  =
  \begin{cases}
  0 & i>0 \\
  k & i=0, \lambda = -w_0\mu
  \end{cases}
  .\end{align*}
\item
  There are only extensions in one specific situation:
  \begin{align*}  
  \operatorname{Ext} _G^i(V(\mu), H^0(\lambda)) = 
  \begin{cases}
  0 & i> 0 \\
  k & i=0, \lambda = \mu
  \end{cases}
  .\end{align*}
\end{enumerate}

\end{theorem}

The following is an important calculation!

\begin{proof}

Step 1: We'll use Frobenius reciprocity twice. We can write the term of
interest in two ways:
\begin{align*}  
H^i(G, H^0(\lambda) \otimes H^0(\mu)) =
H^i(B, H^0(\lambda) \otimes\mu)
\\ \\
H^i(G, H^0(\lambda) \otimes H^0(\mu)) =
H^i(G, \lambda \otimes H^0(\mu))
.\end{align*}

Thus there exists a weight \(\nu\) of \(H^0(\lambda)\) and \(\nu'\) of
\(H^0(\mu)\) such that
\begin{align*}  
\mu + \nu, \lambda + \nu' \in - {\mathbb{N}}\Phi^+ \quad \operatorname{ht}(\mu+\nu), \operatorname{ht}(\lambda + \nu') \leq -i
.\end{align*}

Since \(w_0\lambda\) (resp. \(w_0\mu\)) is the lowest of weight of
\(H_0(\lambda)\) (resp. \(H_0(\mu)\)), it follows that
\begin{align*}  
\mu + w_0 \lambda, \lambda + w_0\mu \in -{\mathbb{N}}\Phi^+
.\end{align*}

Since \(w_0^2 = \operatorname{id}\), we can write
\(\lambda + w_0\mu = w_0(\mu + w_0 \lambda)\). We know that the LHS is
in \(-{\mathbb{N}}\Phi^+\), and the term in parentheses on the RHS is
also in \(-{\mathbb{N}}\Phi^+\). Applying \(w_0\) interchanges
\(\Phi^\pm\), so the RHS is in \({\mathbb{N}}\Phi^+\). But
\({\mathbb{N}}\Phi^+ \cap-{\mathbb{N}}\Phi^+ = \left\{{0}\right\}\),
forcing \(\lambda + w_0 \mu = 0\) and thus \(\lambda = -w_0 \mu\).

Since the height of zero is zero, we have
\begin{align*}  
0 = \operatorname{ht}(\lambda + w_0 \mu)
\leq \operatorname{ht}(\lambda + \nu') \leq -i \implies i=0
.\end{align*}
This shows cohomological vanishing for \(i>0\), the first case in the
theorem statement.

For the remaining case, we can check that
\(H^0(\lambda)^{U} = H^0(\lambda)_{w_0 \lambda}\), and so
\begin{align*}  
\qty{H^0(\lambda) \otimes-w_0 \lambda}^{U^+} = k
.\end{align*}
This shows that \(H^0(B; H^0(\lambda) \otimes-w_0\lambda ) \cong k\),
since
\begin{align*}  
\qty{H^0(\lambda) \otimes-w_0 \lambda}^B = \qty{ \qty{H^0(\lambda) \otimes-w_0 \lambda }^U }^T
.\end{align*}

\end{proof}

\begin{proposition}[?]

Let \(\lambda, \mu \in X(T)_+\) with \(\lambda \not> \mu\). Then we can
calculate the \(i\)th ext by computing the \(i-1\)st: for \(i>0\),
\begin{align*}  
\operatorname{Ext} ^i_G(L(\lambda), L(\mu))
\cong
\operatorname{Ext} ^{i-1}_G(L(\lambda), H^0(\mu) / \mathop{\mathrm{Soc}}_G(H^0(\mu)))
.\end{align*}

\end{proposition}

\begin{remark}

We showed this in a special case. Let \(i=1\) with
\(\lambda \not> \mu\), then
\begin{align*}  
\operatorname{Ext} _G^1(L(\lambda), L(\mu)) \cong
\mathop{\mathrm{Hom}}_G(L(\lambda), H^0(\mu) / \mathop{\mathrm{Soc}}_G(H^0(\mu)))
.\end{align*}
Thus it suffices to understand only the previous layer:

\begin{figure}
\centering
\includegraphics{figures/image_2020-09-30-14-25-09.png}
\caption{Image}
\end{figure}

\end{remark}

\begin{proof}

Consider the SES
\begin{align*}  
0 \to L(\mu) \to H^0(\mu) \to H^0(\mu) / \mathop{\mathrm{Soc}}_G(H^0(\mu)) \to 0
\end{align*}
which yields a LES in homology by applying \(\hom_G(L(\lambda), {-})\).
To obtain the statement, it suffices to show
\(\operatorname{Ext} _G^1(L(\lambda), H^0(\mu)) = 0\) for \(i>0\), since
this is the middle column in the LES.

We can write
\begin{align*}  
\operatorname{Ext} _G^i(L(\lambda), H^0(\mu))
=
H^i(G, L(\lambda) {}^{ \vee }\otimes H^0(\mu)) \quad\text{taking duals} \\
=
H^i(B, L(\lambda) {}^{ \vee }\otimes\mu) \quad\text{by Frobenius reciprocity}
,\end{align*}
so we can obtain a weight \(\sigma\) of
\(L(\lambda) {}^{ \vee }\otimes\mu\) such that
\(\sigma \in - {\mathbb{N}}\Phi^+\) and
\(\operatorname{ht}(-\sigma) \geq i > 0\) by applying the previous
lemma. So \(\sigma = \nu + \mu\) for \(\nu\) some weight of
\(L(\lambda) {}^{ \vee }\).

By rearranging, we find that \(\sigma \in {\mathbb{N}}\Phi^-\). Letting
\(\lambda\) be the lowest weight of \(L(\lambda) {}^{ \vee }\), we find
\(\sigma \geq -\lambda + \mu\) (since this can only lower the weight).

But then \(-\lambda + \mu \in {\mathbb{N}}\Phi^-\), implying
\(-\mu + \lambda \in {\mathbb{N}}\Phi^-\), and the LHS here is equal to
\(\lambda - \mu\). This precisely says \(\lambda > \mu\), which
contradicts the assumption that \(\lambda\) did not dominate \(\mu\). It
may also be the case that \(\lambda = \mu\), which is handled
separately.

\end{proof}

We now want criteria for when we can find the following types of lifts:

\begin{center}
\begin{tikzcd}
 & V \\
L(\lambda) \ar[ur, "\hookrightarrow"] \ar[r, "\hookrightarrow"] & H^0(\lambda) \ar[u, dotted, "\hookrightarrow"]
\end{tikzcd}
\end{center}

\begin{lemma}[Important!]

Let \(V\) be a \(G{\hbox{-}}\)module with
\(0\neq \hom_G(L(\lambda), V)\). If

\begin{itemize}
\item
  \(\hom(L(\mu), V) = 0\),
\item
  \(\operatorname{Ext} _G^1(V(\mu), V) = 0\) for all \(\mu \in X(T)_+\)
  with \(\mu < \lambda\),
\end{itemize}

then \(V\) contains a submodule isomorphic to \(H^0(\lambda)\) and such
a lift/extension exists.

\end{lemma}

\begin{remark}

The ext criterion will be the most important. The idea is to quotient
and continue applying it.

\end{remark}

\begin{proof}

Consider the SES
\begin{align*}  
0 \to L(\lambda) \hookrightarrow V \to V/L(\lambda) \to 0
\end{align*}
as well as
\begin{align*}  
0 \to L(\lambda) \to H^0(\lambda) \to H^0(\lambda)/L(\lambda) \to 0
.\end{align*}

Now want to applying the LES in cohomology by applying
\(\hom_G({-}, V)\), we get a LES of homs over \(G\):
\begin{align*}  
0 &\to \mathop{\mathrm{Hom}}(H^0(\lambda)/L(\lambda), V) \to
\mathop{\mathrm{Hom}}(H^0(\lambda) , V) \to
\mathop{\mathrm{Hom}}(L(\lambda), V)  \\
&\to \operatorname{Ext} ^1(H^0(\lambda)/L(\lambda), V) \to \cdots
.\end{align*}
Thus it suffices to show this \(\operatorname{Ext} ^1\) is zero.

Strategy: show all of the composition factors of
\(H^0(\lambda)/L(\lambda)\) are zero These are all of the form
\(L(\mu)\) for \(\mu < \lambda\), so it now suffices to just show that
\(\operatorname{Ext} _G^1(L(\mu), V) = 0\) when \(\mu < \lambda\).

Observe that we have
\begin{align*}  
0 \to N \to V(\mu) \to L(\mu) \to 0
\end{align*}
where \(N\) are \(L(\sigma)\) composition factors for \(\sigma < \mu\).
So apply \(\hom({-}, V)\):
\begin{align*}  
0 
&\to
\mathop{\mathrm{Hom}}(L(\mu), V) \to
\mathop{\mathrm{Hom}}(V(\mu), V) \to
\mathop{\mathrm{Hom}}(N, V) \\
&\to
\operatorname{Ext} ^1(L(\mu), V) \to
\operatorname{Ext} ^1(V(\mu), V) \to \cdots
.\end{align*}

But we have \(\mathop{\mathrm{Hom}}(N, V) =0\) and
\(\operatorname{Ext} ^1(V(\mu), V) = 0\), which \emph{squeezes} and
forces \(\operatorname{Ext} ^1(L(\mu), V) = 0\).

\end{proof}

Next time: state and prove a cohomological criterion (Donkin, Scott,
proved independently) for a \(G{\hbox{-}}\)module to admit a good
filtration. More about when tensor products of induced modules have good
filtrations.

\hypertarget{friday-october-02}{%
\section{Friday, October 02}\label{friday-october-02}}

Recall that \emph{good filtration} is a chain
\(\left\{{0}\right\} \subseteq V_1 \subseteq \cdots \subseteq V\)
satisfying \(V = \cup V_i\) and \(V_i/V_{i-1} \cong H^0(\lambda_i)\) for
\(\lambda_i\) some weight of \(V\).

\begin{lemma}[?]

Let \(V\) be a \(G{\hbox{-}}\)module and \(\lambda \in X(T)_+\) with
\(\hom_G(L(\lambda), V)\). If \(\hom_G(L(\mu), V) = 0\) for any
\(\mu < \lambda\) and \(\operatorname{Ext} _G^1(V(\mu), V) = 0\) for
\emph{all} \(\mu \in X(T)_+\), then \(V\) contains a submodule
isomorphic to \(H^0(\lambda)\).

\end{lemma}

That is, we have a lift of the following form:

\begin{center}
\begin{tikzcd}
L(\lambda) \ar[d, hook] \ar[r, hook] & V \\
H^0(\lambda) \ar[ru, hook, dotted, "\exists"]
\end{tikzcd}
\end{center}

\begin{theorem}[Cohomological Condition for Good Filtrations]

Let \(V\) be a \(G{\hbox{-}}\)module.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  If \(V\) admits a good filtration, then the number of factors
  isomorphic to \(H^0(\lambda)\), denoted \([V: H^0(\lambda)]\), is
  equal to \(\dim \hom_G(V(\lambda), V)\).
\end{enumerate}

\begin{quote}
Analog of Jordan-Holder. Note that \(H^0(\lambda)\) may not by
irreducible, but changing the filtration can not change the number of
composition factors.
\end{quote}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  Suppose \(\hom_G(V(\lambda), V)<\infty\), then TFAE:
\end{enumerate}

\begin{itemize}
\tightlist
\item
  \(V\) admits a good filtration.
\item
  \(\operatorname{Ext} ^i_G(V(\lambda), V) = 0\) for all
  \(\lambda \in X(T)_+\) and all \(i>0\).
\item
  \(\operatorname{Ext} ^1_G(V(\lambda), V) = 0\) for all
  \(\lambda \in X(T)_+\).
\end{itemize}

\begin{quote}
Much like measuring projectivity: can check all exts, or just the first.
\end{quote}

\end{theorem}

\begin{proof}[Part a]

Suppose \(V\) has a good filtration. Idea: induct on the filtration.

Suppose \(V = H^0(\lambda_1)\), then
\begin{align*}  
[V: H^0(\mu) ] = 
\begin{cases}
0 & \mu \neq \lambda_1 \\
1 & \mu = \lambda_1
\end{cases}
= \dim \hom_G(V(\lambda_1), V)
,\end{align*}
since we know the dimensions of these hom spaces from a previous result.

Suppose now that we have
\begin{align*}  
0 \to H^0(\mu_1) \to V H^0(\mu_2) \to 0
.\end{align*}
Applying \(F \coloneqq\hom_G(V(\lambda), {-})\), we find that
\(\operatorname{Ext} ^1_G\) vanishes. So this leads a SES, and the
dimensions are thus additive. The result follows since \(F\) is
additive.

\end{proof}

\begin{proof}[Part b]

\(1\implies 2\): Use the fact that
\(\operatorname{Ext} ^i_G(V(\lambda), H^0(\mu)) = 0\) for all \(i>0\)
and all \(\mu\).

\(2\implies 3\): Clear!

\(3\implies 1\): Choose a total ordering of weights
\(\lambda_0, \lambda_1, \cdots \in X(T)\) such that if
\(\lambda_i < \lambda_j\) then \(i<j\). Since \(V\neq 0\), there exists
a dominant weight \(\lambda \in X(T)_+\) such that
\(\hom_G(V(\lambda), V) \neq 0\), so choose \(i\) minimally in this
order to produce such a \(\lambda_i\). Idea: use this to start a
filtration.

Then \(\hom(L(\lambda_i), V) \neq 0\), and we have
\begin{align*}  
V(\lambda_i) \twoheadrightarrow L(\lambda_i) \hookrightarrow V
.\end{align*}

We know that
\begin{align*}  
\hom_G(V(\mu), V) = 0 \quad \forall \mu < \lambda_i \\
\hom_G(L(\mu), V) = 0 \quad \forall \mu < \lambda_i \\
\operatorname{Ext} _G^1(L(\mu), V) = 0 \quad \forall \mu \in X(T)_+ \text{ by assumption}
.\end{align*}

So the following map must be an injection, since there is no socle:

\begin{center}
\begin{tikzcd}
          & L(\lambda_i) \ar[r, hook] \ar[d, hook] & V \\
0 \ar[r]  & H^0(\lambda_i) \ar[ur, hook] &
\end{tikzcd}
\end{center}

Set \(V_1 = H^0(\lambda_i)\), so \(V_1 \subseteq V\). We then have a SES
\begin{align*}  
0 \to V_1 \to V \to V/V_1 \to 0 
.\end{align*}

Applying \(\hom(V(\lambda), {-})\) we obtain

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-02-14-26-17.png}
\caption{Cancellation in LES}
\end{figure}

Now iterate this process to obtain a chain
\(V_1 \subseteq V_2 \subseteq \cdots \subseteq V\), and set
\(V' \coloneqq\cup_{i>0} V_i\). Then
\(\dim \hom_G(V(\lambda), V') = \dim \hom_G( V(\lambda), V )\) since
\(\dim \hom_G(V(\lambda), V) < \infty\). But then taking the SES
\begin{align*}  
0\to V' \to V \to V/V' \to 0
\end{align*}
and applying \(\mathop{\mathrm{Hom}}(V(\lambda), {-})\), we have
\(\mathop{\mathrm{Hom}}(V(\lambda), V/V') = 0\) and we get an
isomorphism of homs. But then \(\hom(V(\lambda), V/V') = 0\) for all
\(\lambda \in X(T)_+\), forcing \(V/V'=0\) and \(V=V'\).

\end{proof}

\begin{corollary}[?]

Let \(0\to V_1 \to V \to V_2 \to 0\) be a SES of \(G{\hbox{-}}\)modules
with \(\dim \hom_G(V(\lambda), V_2) < \infty\) for all
\(\lambda \in X(T)_+\). If \(V_1, V\) have good filtrations, then
\(V_2\) also has a good filtration.

\end{corollary}

Note: this is likely difficult to prove without cohomology! But here we
can apply the ext criterion.

\begin{proof}

Let \(\lambda \in X(T)_+\), then

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-02-14-34-18.png}
\caption{Image}
\end{figure}

\end{proof}

For \(\lambda \in X(T)_+\), let \(I(\lambda)\) be the injective hull of
\(L(\lambda)\), so we have
\begin{align*}  
0 \to L(\lambda) \hookrightarrow I(\lambda)
.\end{align*}

\begin{theorem}[?]

Let \(\lambda \in X(T)_+\) and \(I(\lambda)\) be the injective hull of
\(L(\lambda)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(I(\lambda)\) has a good filtration.
\item
  The multiplicity \([I(\lambda): H^0(\mu)]\) is equal to
  \([H^0(\mu): L(\lambda)]\), the composition factor multiplicity.
\end{enumerate}

\begin{quote}
Brauer-Humphreys Reciprocity. Same idea as in category
\({\mathcal{O}}\): multiplicity of Vermas equals multiplicity of
irreducibles.
\end{quote}

\end{theorem}

\begin{proof}[of a]

How to check that it has a good filtration? The cohomological criterion!
So consider \(\operatorname{Ext} ^1_G( V(\sigma), I(\lambda) )\) for all
\(\sigma \in X(T)_+\). We want to show it's zero, but this follows
because \(I(\lambda)\) is injective.

\end{proof}

\begin{proof}[of b]

By the previous result, we have
\begin{align*}  
[I(\lambda): H^0(\mu) ] 
&= \dim \hom_G(V(\mu), I(\lambda)) \\
&= [V(\mu): L(\lambda) ]
.\end{align*}
Why does this second equality hold? The functor
\(\hom_G({-}, I(\lambda))\) is exact, and
\(\hom_G(L(\mu), I(\lambda)) = \delta_{\lambda, \mu}\). If
\(\lambda = \mu\) there's only one morphism, since
\(L(\lambda) \hookrightarrow I(\lambda)\) and
\(\mathop{\mathrm{Soc}}_G I(\lambda) = L(\lambda)\). This means that
they have the same character,
\(\operatorname{ch}H^0(\lambda) = \operatorname{ch}V(\lambda)\), and
this implies that they have the same composition factors.

\end{proof}

\begin{theorem}[Cohomological Criterion for Weyl Filtrations]

Let \(V\) be a \(G{\hbox{-}}\)module.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(V\) admits a Weyl filtration, then
  \begin{align*}
  [V: V(\lambda)] = \dim \hom_G (V, H^0(\lambda))
  \end{align*}
\item
  Suppose that \(\dim \hom_G(V(\lambda), H^0(\lambda)) < \infty\) for
  all \(\lambda \in X(T)_+\). Then TFAE
\end{enumerate}

\begin{itemize}
\tightlist
\item
  \(V\) has a Weyl filtration.
\item
  \(\operatorname{Ext} ^i_G(V, H^0(\lambda)) = 0\) for all
  \(\lambda \in X(T)_+\) and \(i>0\).
\item
  \(\operatorname{Ext} ^1_G(V, H^0(\lambda)) = 0\) for all
  \(\lambda \in X(T)_+\).
\end{itemize}

\end{theorem}

\hypertarget{monday-october-05}{%
\section{Monday, October 05}\label{monday-october-05}}

Crelle 1988 (CPS: Cline Parshall Scott)

Let HWC denote a highest weight category.

\begin{example}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  BGG Category \({\mathcal{O}}\)
\item
  \(\operatorname{Rat}(G)\) for \(G\) a reductive algebraic group
\item
  \(\mathsf{Perv}_W(G/B) \cong {\mathcal{O}}_0\)
\end{enumerate}

\end{example}

See

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Donkin: On generalized Schur algebras
\item
  Irving: BGG algebras
\end{enumerate}

There is a equivalence between HWC and QHA (quasi-hereditary algebras).

\begin{remark}

Key Points

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(L(\lambda) = \mathop{\mathrm{Soc}}_G \nabla(\lambda)\) and
  \(\nabla(\lambda) = A(\lambda)\).
\item
  All composition factors of \(\nabla(\lambda)\) satisfy
  \(\mu \leq \lambda\)
\item
  We have cohomological vanishing:
  \begin{align*}
  \operatorname{Ext} _G^i(\Delta(\lambda), \nabla(\mu)) = 
  \begin{cases}
  0 & i >0 \\
  0 & i=0, \lambda \neq \mu \\
  k & i=0. \lambda = 0
  \end{cases}
  \end{align*}
\end{enumerate}

\end{remark}

Interval finite poset: we'll have a cone \(\Lambda\) of positive
weights:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-05-14-14-30.png}
\caption{Image}
\end{figure}

\begin{quote}
See handout!
\end{quote}

\begin{theorem}[?]

Let \(G ,G'\) be rational \(G{\hbox{-}}\)modules admitting good
filtrations. Then the tensor product \(V\otimes V'\) also admits a good
filtration.

\end{theorem}

\begin{itemize}
\tightlist
\item
  First proofs:

  \begin{itemize}
  \tightlist
  \item
    JP Wong, Type A
  \item
    Donkin, all but characteristic 2 and \(E_7, E_8\).
  \item
    O. Mathieu, general proof using algebraic geometry
  \end{itemize}
\end{itemize}

\begin{example}

Let \(G = {\operatorname{SL}}(n, k)\) and take the natural
representation \(V = H^0(w_1)\). Then \(V^{\otimes d}\) has a good
filtration.

\end{example}

\begin{theorem}[?]

Let \(J\subset \Delta\) be a subset of simple roots. If
\(V \in {\mathsf{Mod}}(G)\) has a good filtration and \(L_J\) is a Levi
factor, then \(V{\downarrow_{L_J}}\) has a good filtration.

\end{theorem}

\begin{theorem}[?]

Let \({\mathfrak{g}}= \operatorname{Lie}(G)\) and \(p\) be a \emph{good
prime} (doesn't divide any of the coefficients of the highest weight).
Then the symmetric algebra \(S({\mathfrak{g}})\) has a good filtration.

\end{theorem}

\begin{remark}

For \(p\geq 3(h-1)\), the exterior algebra \(\Lambda({\mathfrak{g}})\)
also admits a good filtration. Question: Is this true for all primes
\(p\)? Or potentially for all \emph{good} primes \(p\)?

\end{remark}

\hypertarget{polynomial-representation-theory}{%
\subsection{Polynomial Representation
Theory}\label{polynomial-representation-theory}}

Let \(G = \operatorname{GL}(n, k)\), then a module for \(G\) is
\textbf{polynomial} iff the weights
\(\lambda = (\lambda_1, \cdots, \lambda_n)\) satisfy
\(\lambda_j \geq 0\) for all \(j\).

\begin{example}

For \(V\) the natural representation, the weights are the unit vectors
\({\varepsilon}_1, \cdots, {\varepsilon}_n\), so \(V\) is a polynomial
representation. Then \(V^{\otimes d}\) is again polynomial by a previous
remark.

\end{example}

\begin{remark}

Note that the adjoint representation
\({\mathfrak{g}}\cong V\otimes V {}^{ \vee }\) is not a polynomial
representation.

\end{remark}

\begin{theorem}[?]

There is an equivalence
\begin{align*}  
\mathrm{Poly}(G) \cong \bigoplus_{j\geq 0} {\mathsf{Mod}}(S(n, d))
,\end{align*}
where this Schur algebra \(S(n, d)\) is given by
\(\mathop{\mathrm{End}}_{\Sigma_d}(V^{\otimes d})\) where \(\Sigma_d\)
is the symmetric group of \(d\) letters.

The theorem is that \({\mathsf{Mod}}(S(n, d))\) is a QHA, and thus a
highest weight category.

\end{theorem}

\begin{remark}

This is a finite-dimensional algebra, so we should be able to calculate
the dimensions, index by highest weights, write the standard/costandard
modules, etc. There is a correspondence
\begin{align*} 
\left\{{\substack{\text{Simple modules for }S(n, d)}}\right\}
\iff
\left\{{\substack{\Lambda^+(n, d) \text{ partitions of $d$ with at most $n$ parts}}}\right\}
.\end{align*}

We can compute
\begin{align*}  
\dim S(n, d) = {n^2 + d - 1 \choose n^2 - 1}
,\end{align*}
and simple modules correspond to \(L(\lambda)\) for
\(\operatorname{GL}_n\) where \(\lambda\) is a polynomial
representation.

\end{remark}

\begin{theorem}[?]

\(S(n, d)\) is semisimple if and only if

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(k = {\mathbb{C}}\) or characteristic zero, or
\item
  \(d < p\).
\end{enumerate}

\begin{quote}
For latter condition, see Maschke's theorem
\end{quote}

\end{theorem}

\begin{example}

Consider \(S(2, 3)\) for \(p=2\), so \(G = \operatorname{GL}(2)\). Then
\begin{align*}  
\dim S(2, 3) = {4+3-1 \choose 3} = {6\choose 3} = 20
.\end{align*}

The only admissible partitions are thus

\begin{itemize}
\tightlist
\item
  \((3)\), and
\item
  \((2, 1)\).
\end{itemize}

Then \(L(2, 1) = L("w")\) as an
\({\operatorname{SL}}(2){\hbox{-}}\)module, so
\begin{align*}
\dim L(2, 1) = 2
\end{align*}
Then \(L(3, 0) = L("3w")\) as an
\({\operatorname{SL}}(2){\hbox{-}}\)module. We can compute
\begin{align*}  
L(3) = L(1, 0)^{(1)} \otimes L(1, 0)
,\end{align*}
and since each is 2-dimensional, we get \(\dim L(3) = 4^2 + 2^2 = 20\).

Note that the sum of the squares of the dimensions of the irreducibles
are equal to the total dimension, which shows this module is semisimple.
But this contradicts the theorem! So it turns out there is a third
condition, namely this exact case.

\end{example}

Next time: look at structure of injective modules, then the theory of
Bott-Borel-Weil for higher sheaf cohomology.

\hypertarget{wednesday-october-07}{%
\section{Wednesday, October 07}\label{wednesday-october-07}}

\hypertarget{schur-algebras}{%
\subsection{Schur Algebras}\label{schur-algebras}}

Let \(G = \operatorname{GL}(n, k)\), then polynomial representations of
\(G\) are equivalent to \(S(n, d)\) modules for all \(d\geq 0\), where
we can note that
\(S(n, d) = \mathop{\mathrm{End}}_{\Sigma_d}(V^{\otimes d})\). We'll
have a correspondence
\begin{align*}  
\left\{{\substack{L(\lambda) \text{ simple modules for } S(n,d)}}\right\}
\iff
\Lambda^+(n, d) \text{, partitions of $d$ with at most $n$ parts}
,\end{align*}

\begin{example}

\begin{quote}
Good example, can see all filtrations at work, tilting modules, etc.
\end{quote}

Consider \(S(3, 3)\) for \(p=3\), we then have the partitions
\(\Lambda^+(3, 3) = \left\{{(3), (2, 1), (1,1,1)}\right\}\). We can
think of these in the \({\varepsilon}\) basis as
\((3) = (3,0,0), (2,1) = (2,1,0)\). Since
\({\operatorname{SL}}(3, k) \subset \operatorname{GL}(3, k)\), we can
find the \(SL(3, k)\) weights by taking successive differences to yield
\((3, 0), (1, 1), (0, 0)\) with the corresponding picture

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-00-10.png}
\caption{Image}
\end{figure}

We can compute

\begin{itemize}
\tightlist
\item
  \(L(1,1,1) = H^0(1,1,1)\)
\item
  \(L(2, 1) = H^0(2, 1)\)
\item
  \(L(3) = H^0(3)\)
\end{itemize}

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-02-04.png}
\caption{Image}
\end{figure}

We have a form of Brauer reciprocity:
\begin{align*}  
[I(\lambda): H^0(\mu)] = [H^0(\mu) : L(\lambda) ] 
.\end{align*}

We can now compute the injective hulls:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-05-28.png}
\caption{Image}
\end{figure}

What are the tilting modules? We can use the fact that
\(L(1^3) = V(1^3)\). It has a good filtration and a Weyl filtration and
thus must be the tilting module for \(L(1^3)\).

Using the following fact:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-07-57.png}
\caption{Image}
\end{figure}

We can compute the following:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-10-44.png}
\caption{Image}
\end{figure}

\end{example}

\hypertarget{simplicity-of-h0lambda}{%
\subsection{\texorpdfstring{Simplicity of
\(H^0(\lambda)\)}{Simplicity of H\^{}0(\textbackslash lambda)}}\label{simplicity-of-h0lambda}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(k = {\mathbb{C}}\) implies \(L(\lambda) = H^0(\lambda)\) for all
  \(\lambda \in X(T)_+\)
\item
  \(k= \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\)
  implies \(L(\lambda) = H^0(\lambda)\) if
  \({\left\langle {\lambda},~{\alpha_0 {}^{ \vee }} \right\rangle} \leq 1\)
  where \(\alpha_0\) is the highest short root.
\end{enumerate}

Such \(\lambda\) are referred to as \emph{minuscule weights}.

\begin{example}

For type \(A_n\), we have \(\alpha_0 = \sum_{i=1}^n \alpha_i\). For type
\(G_2\), we have
\(\alpha_0 {}^{ \vee }= 2\alpha_1 {}^{ \vee }+ 3\alpha_2 {}^{ \vee }\).

\end{example}

\begin{example}

In type \(A_n\), set \(\lambda = \sum_{j=1}^n c_j w_j\) where
\(c_j \geq 0\). Then
\({\left\langle {\lambda},~{\alpha_0 {}^{ \vee }} \right\rangle} = \sum c_j \leq 1\),
so \(\lambda\) is minuscule iff \(\lambda = 0\) or \(\lambda = w_j\) for
some \(j\).

\end{example}

\begin{remark}

Quick timeline:

\begin{itemize}
\tightlist
\item
  2015, Cantrell lectures by Dick Gross at UGA
\item
  Fall 2015: email to Dan Nakano from Skip Garibaldi, conjecture from
  Gross without a proof
\end{itemize}

\begin{proposition}[Gross]

The simple module is equal to the induced module, so
\(L(\lambda) = H^0(\lambda)\), for all \(p\) iff \(\lambda\) is
minuscule, or if \(L(\lambda) = {\mathfrak{g}}\) for \(\Phi = E_8\).

\end{proposition}

\begin{itemize}
\tightlist
\item
  Proved by Garibaldi-Nakano-Guralnick, appeared in Journal of Algebra
\end{itemize}

\end{remark}

\hypertarget{bott-borel-weil-theorem}{%
\subsection{Bott-Borel-Weil Theorem}\label{bott-borel-weil-theorem}}

We can consider the higher right-derived functors of \(\lambda\), given
by \(H^i(\lambda) = R^i \mathop{\mathrm{ind}}_B^G \lambda\) for
\(\lambda \in X(T)\). You can think of this as the higher sheaf
cohomology of the flag variety,
\(\mathcal{H}^i(G/B, \mathcal{L}(\lambda))\).

We have \textbf{Kempf Vanishing}: \(H^i(\lambda) = 0\) for all \(i>0\)
when \(\lambda \in X(T)_+\) is dominant (although other things may
happen for non-dominant weights). There is a correspondence
\((G, T) \iff (W, \Phi)\), and since \(W\) is generated by simple
reflections, we can write any \(w\in W\) as \(w=\prod s_{\alpha_i}\). A
\emph{reduced expression} is one in which the length can not be
shortened, and any two reduced expressions necessarily have the same
length (number of simple reflections).

\begin{example}

For \(\Phi = A_2\), we have
\(w_0 = s_{\alpha_1} s_{\alpha_2} s_{\alpha_1} = s_{\alpha_2} s_{\alpha_1} s_{\alpha_2}\).

\end{example}

\hypertarget{dot-action-on-weights}{%
\subsubsection{Dot Action on Weights}\label{dot-action-on-weights}}

We can let \(W\) act on \(X(T)\) by reflections by the formula
\(s_\alpha \lambda = \lambda - {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}\alpha\).
We then shift the action by setting
\(s_\alpha \cdot \lambda = w(\lambda+\rho)-\rho\) where
\(\rho = {1\over 2} \sum_{\alpha\in \Phi^+} \alpha = \sum_{j=1}^n w_j\).

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-33-00.png}
\caption{Image}
\end{figure}

\begin{theorem}[Bott-Borel-Weil]

Let \(G\) be a reductive algebraic group and \(k={\mathbb{C}}\). For
\(\lambda \in X(T)_+\), we can describe the sheaf cohomology:
\begin{align*}  
\mathcal{H}^i(w\cdot \lambda)
=
\begin{cases}
H^0(\lambda) & i=\ell(w) \\
0 & \text{otherwise}
\end{cases}
.\end{align*}

Moreover, if \(\lambda \not\in X(T)_+\) and
\({\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\)
for all \(\alpha \in \Delta\), then
\(\mathcal{H}^i(w\cdot \lambda) = 0\) for all \(w\in W\).

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-41-58.png}
\caption{Image}
\end{figure}

\end{theorem}

Wide open in characteristic \(p\), can say some things. We'll prove this
in characteristic zero.

Recall that \(k={\mathbb{C}}\) and \(H^0(\lambda) = L(\lambda)\). We'll
want to reduce to \({\operatorname{SL}}(2, {\mathbb{C}})\) parabolics.
For \(\alpha\in\Delta\), let \(P_\alpha\) be the associated parabolic
\(P_\alpha = L_\alpha \rtimes U_\alpha\), which is parabolic of type
\(A_1\).

Idea: \(\alpha\) generates an \({\operatorname{SL}}_2\) subgroup (the
Levi factor), like the Borel but sticks out in one dimension:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-07-14-47-17.png}
\caption{Image}
\end{figure}

Then
\begin{align*} 
s_\alpha \cdot \lambda = s_\alpha(\lambda + \rho) - \rho \\
= \lambda + \rho - {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle}\alpha - \rho \\
= \lambda - {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle}\alpha
.\end{align*}

Next time: proof of Bott-Borel-Weil and its generalization to
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\).
For \(B\subset P_\alpha \subset G\), we'll have a spectral sequence
\begin{align*}  
E_2^{i, j} = R^i \mathop{\mathrm{ind}}_{P_\alpha}^G R^j \mathop{\mathrm{ind}}_B^{P_\alpha}  \Rightarrow R^{i+j} \mathop{\mathrm{ind}}_B^G  \lambda = H^{i+j}(\lambda)
.\end{align*}

\hypertarget{friday-october-09}{%
\section{Friday, October 09}\label{friday-october-09}}

Last time: Bott-Borel-Weil. Stated for characteristic zero, working
toward a generalization.

Let \(\Delta\) be the set of simple roots, and \(\alpha\in \Delta\). We
can form a Levi decomposition
\(P_\alpha \coloneqq L_\alpha \rtimes U_\alpha\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-09-13-58-02.png}
\caption{Image}
\end{figure}

We have \(B \subseteq P_\alpha \subseteq G\). The dot action is given by
the following: Let \(W\) be the Weyl group, then \(W\) acts on \(X(T)\)
by \(w\cdot \lambda = w(\lambda + \rho) - \rho\), where
\begin{align*}  
\rho = {1\over 2} \sum_{\alpha\in \Phi^+} \alpha = \sum_{i=1}^n w_n
.\end{align*}

We obtained a formula
\begin{align*}  
S_\alpha \cdot \lambda = \lambda - {\left\langle {\lambda  + \rho},~{\alpha {}^{ \vee }} \right\rangle} \alpha
.\end{align*}

\hypertarget{bott-borel-weil-theory}{%
\subsection{Bott-Borel-Weil Theory}\label{bott-borel-weil-theory}}

\begin{proposition}[?]

Let \(\alpha\in\Delta\) be simple and \(\lambda \in X(T)\) be an
arbitrary weight. Then

\begin{itemize}
\item
  \(U_\alpha\) acts trivially on
  \(\mathop{\mathrm{ind}}_B^{P_\alpha} \lambda\).
\item
  (Kempf's Vanishing for \(P_\alpha\)) If
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = r \geq 0\),
  then
  \begin{align*}  
  R^i \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0 \qquad \text{for } i \geq 0
  ,\end{align*}
  and \(\dim \mathop{\mathrm{ind}}_B^{P_\alpha}\lambda = r + 1\).
\item
  If
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = -1\),
  then \(R^i \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0\) for all
  \(i\).
\item
  If
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \leq -2\),
  then

  \begin{itemize}
  \item
    \(R^i \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0\) for
    \(i \neq 1\), and
  \item
    \(\dim R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = r+1\)
  \end{itemize}
\end{itemize}

Note: we have
\begin{align*}  
\mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = S^r(V) \qquad &\text{when } {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = r \geq 0 \\
R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} = S^r(V) {}^{ \vee }\qquad&\text{where $V$ is a 2-dim representation and } {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \leq -2 \\
&\text{and } r = {\left\lvert {{\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}} \right\rvert} - 1
.\end{align*}

\end{proposition}

This gives us an analog of \(A_1\) or \({\operatorname{SL}}_2\) theory.
Also note that we have Serre duality:
\begin{align*}  
H^1(\lambda) = H^0( - (\lambda + 2\rho) ) {}^{ \vee }
.\end{align*}

\begin{corollary}[?]

Let \(\alpha\in \Delta\) and \(\lambda\in X(T)\), and suppose
\(\lambda\) is dominant with respect to \(\alpha\),
i.e.~\({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\).

\begin{itemize}
\item
  If \(\operatorname{ch}(k) = 0\) then
  \(\mathop{\mathrm{ind}}_B^{P_\alpha}\lambda = R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} s_\alpha \cdot \lambda\)
\item
  If \(\operatorname{ch}(k) = p\) and if there exists an \(s, m\) with
  \(0<s<p\) and
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = sp^m - 1\)
  (Steinberg weights), then
  \begin{align*}  
  \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = R^1 \mathop{\mathrm{Ind}}_B^{P_\alpha} s_\alpha \cdot \lambda
  .\end{align*}
\end{itemize}

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-09-14-14-39.png}
\caption{O}
\end{figure}

\end{corollary}

The proof of this will use a Grothendieck-type spectral sequence of the
form
\begin{align*}  
E_2^{i, j} = R^i \mathop{\mathrm{ind}}_{P_\alpha}^G \qty{ R^j \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda} \Rightarrow R^{i+j} \mathop{\mathrm{ind}}_B^G \lambda
.\end{align*}

We'll have a version of \emph{Grothendieck vanishing}:
\begin{align*}  
R^j \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0 \qquad\text{for } j > \dim P_\alpha/B = 1
.\end{align*}

So the resulting spectral sequence will only be supported on the first
two lines, and \(E_3 = E_\infty\). Note the differential will be of
bidegree \({\partial}_r \leadsto (r, 1-r)\), and \(E_2\) will look like
the following,

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-09-14-30-47.png}
\caption{Image}
\end{figure}

Recall that
\(R^i \mathop{\mathrm{ind}}_B^G \lambda \coloneqq H^i(\lambda)\)

\begin{proposition}[?]

Let \(\alpha\in\Delta\) and \(\lambda \in X(T)\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  If
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = -1\),
  then \(H^{-}(\lambda) = 0\).
\item
  If
  \({\left\langle {\lambda},~{ \alpha {}^{ \vee }} \right\rangle} \geq 0\),
  then \(H^i(\lambda) = R^i \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda\)
  for all \(i\geq 0\).
\item
  If
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \leq -2\),
  then
  \begin{align*}  
  H^i(\lambda) = R^{i-1} \mathop{\mathrm{ind}}_{P_\alpha}^G \qty{ R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda } \qquad \forall i
  .\end{align*}
\item
  Suppose
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\).
  If \(\operatorname{ch}(k) = 0\), or \(\operatorname{ch}(k) = p> 0\)
  and
  \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = sp^n - 1\),
  then
  \begin{align*}  
    H^i(\lambda) = H^{i+1}(s_\alpha\cdot \lambda)
    .\end{align*}
\end{enumerate}

\end{proposition}

\begin{proof}[of a]

If
\({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = -1\),
then \(R^{-}\mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0\). But this
is what appears as the ``coefficients'' in the spectral sequence, so
\(E_2^{{-}, {-}} = 0\) and this
\(R^{-}\mathop{\mathrm{ind}}_B^{P_\alpha} = 0\).

\end{proof}

\begin{proof}[of b]

If \({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = 0\),
then \(R^j \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0\) for all
\(j>0\). Thus only the bottom line survives, and the spectral sequence
degenerates on page 2. Thus
\(E_2^{1, 0} = R^i \mathop{\mathrm{ind}}_B^G \lambda\), where the LHS is
equal to
\(R^i \mathop{\mathrm{ind}}_{P_\alpha}^G \qty{\mathop{\mathrm{ind}}_B^{P_\alpha} \lambda }\).

\end{proof}

\begin{proof}[of c]

If
\({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} = -2\),
then \(R^i \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = 0\) for
\(i\neq 1\), so only \(i=1\) survives Then
\begin{align*}
R^{i-1} \mathop{\mathrm{ind}}_{P_\alpha}^G \qty{ \mathop{\mathrm{ind}}_B^{PP_\alpha} \alpha} = R^i \mathop{\mathrm{ind}}_B^G \lambda
,\end{align*}
so there is some dimension shifting.

\end{proof}

\begin{proof}[of d]

If
\({\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\),
then by (b),
\begin{align*}  
H^i(\lambda) 
&= R^i \mathop{\mathrm{ind}}_{P_\alpha}^G \qty{ \mathop{\mathrm{ind}}_B^{P_\alpha} \lambda } && \text{by c}\\
&= R^i \mathop{\mathrm{ind}}_{P_\alpha}^G \qty{ R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} s_\alpha\cdot \lambda } && \text{by corollary}\\
&= H^{i+1}(s_\alpha\cdot \lambda)
.\end{align*}

We can then check that
\begin{align*}  
s_\alpha \cdot \lambda
&= \lambda - {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle}\alpha \\
&= \lambda - \qty{ {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} + 1 }\alpha && \text{using } {\left\langle {\rho},~{\alpha {}^{ \vee }} \right\rangle} = 1 \\ \\
\implies 
{\left\langle {s_\alpha \cdot \lambda},~{\alpha {}^{ \vee }} \right\rangle}
&= {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} - \qty{ {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}+1 }{\left\langle {\alpha},~{\alpha {}^{ \vee }} \right\rangle} \\
&= {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} - \qty{ {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle}+1 }2 \\
&= -{\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} - 2 \\
&\leq -2
.\end{align*}

\end{proof}

Now define
\begin{align*}  
\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{{\mathbb{Z}}} 
&\coloneqq
\left\{{
\lambda \in X(T) {~\mathrel{\Big|}~}0 \leq {\left\langle {\lambda+\rho},~{\beta {}^{ \vee }} \right\rangle} \,\forall \beta \in \Phi^+
}\right\} \qquad\text{ if } \operatorname{ch}(k) = 0 \\
&\coloneqq
\left\{{
\lambda \in X(T) {~\mathrel{\Big|}~}0 \leq {\left\langle {\lambda+\rho},~{\beta {}^{ \vee }} \right\rangle} \leq \operatorname{ch}(k) \,\forall \beta \in \Phi^+
}\right\} \qquad\text{if } \operatorname{ch}(k) = p
.\end{align*}

Idea:

\includegraphics{figures/image_2020-10-09-14-45-08.png}
\includegraphics{figures/image_2020-10-09-14-45-20.png}

\begin{theorem}[Bott-Borel-Weil Generalization, due to Andersen]

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If
  \(\lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
  and \(\lambda \not\in X(T)_+\), then \(H^0(w\cdot \lambda) = 0\).
\item
  If
  \(\lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\cap X(T)_+\),
  then for all \(w\in W\),
  \begin{align*}  
  H^i(w\cdot \lambda) = 
  \begin{cases}
  H^0(\lambda) & i= \ell(w) \\
  0 & \text{otherwise}
  \end{cases}
  .\end{align*}
\end{enumerate}

\end{theorem}

Note that this covers everything in the \(\operatorname{ch}(k) = 0\)
case, but only gives the following hexagon in the
\(\operatorname{ch}(k) = p\) case:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-09-14-48-41.png}
\caption{Image}
\end{figure}

\begin{remark}

\textbf{Open Problem}: Determine \(\operatorname{ch}H^i(\lambda)\) for
\(\lambda\in X(T)\) in characteristic \(p>0\).

Andersen provided necessary an sufficient conditions for
\(H^1(\lambda) \neq 0\) and computed
\(\mathop{\mathrm{Soc}}_G H^1(\lambda)\).

\end{remark}

\hypertarget{monday-october-12}{%
\section{Monday, October 12}\label{monday-october-12}}

\hypertarget{proof-of-bott-borel-weil}{%
\subsection{Proof of Bott-Borel-Weil}\label{proof-of-bott-borel-weil}}

Recall the Bott-Borel-Weil theorem: in characteristic zero, we're
looking at the closure of the region containing the fundamental region
\(C_{\mathbb{Z}}\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-12-13-58-45.png}
\caption{Image}
\end{figure}

\begin{theorem}[due to Aandersen]

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If
  \(\lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
  and \(\lambda \not\in X(T)_+\) then \(H^0(w\circ \lambda) = 0\).
\item
  If
  \(\lambda\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\cap X(T)_+\)
  then for all \(w\in W\), we have
  \begin{align*}  
  H^i(w\cdot \lambda) = 
  \begin{cases}
  H^0(\lambda)& i = \ell(w) \\
  0 & \text{otherwise}
  \end{cases}
  .\end{align*}
\end{enumerate}

\end{theorem}

\begin{proof}[of a]

For (a): we use induction on \(\ell(w)\). For \(\ell(w) = 0\), we have
\(w = \operatorname{id}\). Let
\(\lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
and \(\lambda\not\in X(T)_+\). Then
\begin{align*}  
0 
&\leq {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \\
&= {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} + 1 \\
\implies {\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} &= -1
.\end{align*}
Applying the previous proposition, we get \(H^0(\lambda) = 0\).

\end{proof}

\begin{proof}[of b]

For the base case \(w=\operatorname{id}\), this follows from Kempf
vanishing. Assuming the result holds for any word of length
\(l<\ell(w)\), if \(\ell(w) > 0\), there exists some simple reflection
\(s_\alpha\) for \(\alpha\in\Delta\) such that
\(\ell(s_\alpha w) = \ell(w) - 1\). Moreover,
\(w^{-1}(\alpha) \in -\Phi^+\), so set
\(\beta = -w^{-1}(\alpha) \in \Phi^+\). We can the make the following
computation:
\begin{align*}  
{\left\langle {(s_\alpha w) \cdot \lambda},~{\alpha {}^{ \vee }} \right\rangle}
&= {\left\langle {(s_\alpha w)(\lambda+\rho) - \rho},~{\alpha {}^{ \vee }} \right\rangle}  \\
&= {\left\langle {(s_\alpha w)(\lambda+\rho)},~{\alpha {}^{ \vee }} \right\rangle} - 1 \\
&= {\left\langle {w(\lambda+\rho)},~{s_\alpha \alpha {}^{ \vee }} \right\rangle} - 1 \\
&= - {\left\langle {w(\lambda+\rho)},~{\alpha {}^{ \vee }} \right\rangle} - 1  \\
&= {\left\langle {\lambda + \rho},~{-w^{-1}\alpha {}^{ \vee }} \right\rangle} - 1 \\
&= {\left\langle {\lambda + \rho},~{\beta {}^{ \vee }} \right\rangle} - 1 \\
&\geq -1
\end{align*}
and
\({\left\langle {(s_\alpha w)\cdot \lambda},~{ \alpha {}^{ \vee }} \right\rangle} < \rho\)
since
\(\lambda\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\).
Note that we've used the fact that the inner product is
\(W{\hbox{-}}\)invariant.\\

Now if
\({\left\langle {(s_\alpha w)\cdot \lambda},~{ \alpha {}^{ \vee }} \right\rangle} \geq 0\),
we can apply the prior proposition part (d). Here we use the fact that
\(\mathop{\mathrm{ind}}_B^{P_\alpha}(s_\alpha w)\lambda\) is simple.
Applying the inductive hypothesis yields
\begin{align*}  
H^i(s_\alpha - \lambda) = H^{i+1}(w\cdot \lambda)
.\end{align*}

Now if
\({\left\langle {s_\alpha w \cdot \lambda},~{\alpha {}^{ \vee }} \right\rangle} = -1\),
then
\begin{align*}  
-1 &= {\left\langle {\lambda + \rho},~{\beta {}^{ \vee }} \right\rangle} - 1 \\
\implies {\left\langle {\lambda + \rho},~{\beta {}^{ \vee }} \right\rangle} &= 0 \\
\implies {\left\langle {\lambda},~{\beta {}^{ \vee }} \right\rangle} &= 0 \\
& \cdots
.\end{align*}

\todo[inline]{Missing computation}

Then applying (a) yields \(H^1(w\cdot \lambda) = 0\).

\end{proof}

\hypertarget{serre-duality-and-grothendieck-vanishing}{%
\subsection{Serre Duality and Grothendieck
Vanishing}\label{serre-duality-and-grothendieck-vanishing}}

Let \(P\) be a parabolic subgroup,
i.e.~\(P_J = P \coloneqq L_J \rtimes U_J\) for some
\(J\subseteq \Delta\). Set
\(n(P) = {\left\lvert {\Phi^+} \right\rvert} - {\left\lvert {\Phi^+_J} \right\rvert}\).

\begin{example}

Let \(\Phi = A_4\), which has ten simple roots:

\begin{itemize}
\tightlist
\item
  \(\alpha_i, 1\leq i \leq 4\)
\item
  \(\alpha_i + \alpha_{i+1}\), \(i=1,2,3\).
\item
  \(\alpha_1 + \alpha_2 +\alpha_3\), \(\alpha_2 + \alpha_3 + \alpha_4\)
\item
  \(\sum_{i=1}^4 \alpha_i\).
\end{itemize}

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-12-14-21-20.png}
\caption{Image}
\end{figure}

Then \(n(P) = 10 - 3 = 7\).

\end{example}

\begin{theorem}[Grothendieck Vanishing]

\begin{align*}  
R^i \mathop{\mathrm{ind}}_P^G M = 0 \qquad \text{for } i > n(P)
.\end{align*}

\end{theorem}

\begin{theorem}[Serre Duality]

\begin{align*}  
\qty{ R^i \mathop{\mathrm{ind}}_B^G M } {}^{ \vee }\cong R^{n(P) -i} \mathop{\mathrm{ind}}_P^G M {}^{ \vee }\otimes(-2\rho_P)
.\end{align*}
where
\begin{align*}
\rho_p \coloneqq{1\over 2}\sum_{\beta \in \Phi^+ \setminus\Phi_J} \beta
\end{align*}

\end{theorem}

\begin{example}

Take \(B = P\) and \(M = \lambda\). Then
\(\lambda {}^{ \vee }= -\lambda\), so
\begin{align*}  
\qty{ R^i \mathop{\mathrm{ind}}_B^G \lambda } {}^{ \vee }\cong R^{{\left\lvert {\Phi^+} \right\rvert} -i} \mathop{\mathrm{ind}}_P^G (- \lambda)  {}^{ \vee }\otimes(-2\rho)
.\end{align*}
From this we can conclude
\begin{align*}  
H^i(\lambda) = H^{n-i} (-\lambda - 2\rho) {}^{ \vee }
,\end{align*}
where \(n = {\left\lvert {\Phi^+} \right\rvert}\).

\end{example}

\begin{corollary}[?]

Let
\(\lambda \in X(T)_+ \cap\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
be a dominant weight. Then

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  The irreducible representations are given by
  \(L(\lambda) = H^0(\lambda)\).
\item
  \(\operatorname{Ext} _G^1(L(\lambda), L(\mu)) = 0\) for all
  \(\lambda, \mu\) in
  \(\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\).
\item
  If \(\operatorname{ch}(k) = 0\), so
  \(X(T)_+ \subset \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\),
  then all \(G{\hbox{-}}\)modules are completely reducible.
\end{enumerate}

\end{corollary}

\begin{proof}[of a]

Note that the longest element takes positive roots to negative roots, so
\(w_0 \rho = - \rho\), and moreover
\(-w_0(\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}) = \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\).
We also have
\begin{align*}
w_0 \cdot ( w_0 \lambda) 
&= w_0 (-w_0 \lambda + \rho) - \rho \\
&= -\lambda + w_0 \rho - \rho \\
&= -\lambda - 2\rho
.\end{align*}
By Serre duality, if we take the Weyl module we obtain
\begin{align*}
V(-w_0 \lambda) 
&\coloneqq H^0(\lambda) {}^{ \vee }\\
&= H^n(-\lambda - 2\rho) \\
&= H^n(w_0 \cdot (-w_0 \lambda)) \\
&= H^n(-w_0 \lambda) \qquad\text{by Bott-Borel-Weil}
,\end{align*}
where we've used that
\(\ell(w_0) = {\left\lvert {\Phi^+} \right\rvert}\). We know that
\(L(-w_0 \lambda) \subseteq \mathop{\mathrm{Soc}}H^0(-w_0 \lambda) = V(-w_0 \lambda) \twoheadrightarrow L(-w_0 \lambda)\),
where the last term is contained in the head. But this means that this
splits, so by indecomposability we must have
\(L(-w_0 \lambda) = H^0(-w_0 \lambda) = V(-w_0 \lambda)\). So we can
conclude
\begin{align*}  
L(\mu) = H^0(\mu) = V(\mu) \qquad \forall \mu \in X(T)_+ \cap\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}
.\end{align*}

\end{proof}

\begin{proof}[of b and c]

Suppose \(\operatorname{Ext} _G^1(L(\lambda), L(\mu)) \neq 0\), then
\(L(\lambda)\) is in \(H^0(\mu) / \mathop{\mathrm{Soc}}_G H^0(\mu) = 0\)
and \(L(\mu)\) is in
\(H^0(\lambda) / \mathop{\mathrm{Soc}}_G H^0(\lambda) = 0\), but this
forces \(\operatorname{Ext} _G^1(L(\lambda), L(\mu)) = 0\).\\

Part (c) follows from part (b).

\end{proof}

\hypertarget{weyls-character-formula}{%
\subsection{Weyl's Character Formula}\label{weyls-character-formula}}

Problem: Determine \(\operatorname{ch}H^0 \lambda\) for
\(\lambda \in X(T)_+\).

Solution: Let
\(A(\lambda) = \sum_{w\in W} \operatorname{sgn}(w) e^{w\lambda} \in {\mathbb{Z}}[X(T)]\),
where we sum over the usual Weyl group and not the affine Weyl groups,
taken as a formal sum in the group algebra on the weight lattice. We can
then state Weyl's character formula:
\begin{align*}  
\operatorname{ch}H^0(\lambda) = {A(\lambda + \rho) \over A(\rho)} \qquad \text{for }\lambda \in X(T)_+
.\end{align*}
This is a formal sum, so it's surprising that the bottom term even
divides the top. But there is a great deal of cancellation, we'll see
this in examples such as \(\operatorname{GL}_3\).

\hypertarget{formal-characters}{%
\subsubsection{Formal Characters}\label{formal-characters}}

Let \(M\) be a \(T{\hbox{-}}\)module, then define the \emph{character}
\begin{align*}  
\operatorname{ch}M\coloneqq\sum_{\mu\in X(T)} \qty{\dim M_\mu} e^\mu \quad \in {\mathbb{Z}}[X(T)]
.\end{align*}

We then define the \emph{Euler characteristic}
\begin{align*}  
\chi(M) \coloneqq\sum_{i\geq 0} (-1)^i \operatorname{ch}H^i(M)
.\end{align*}
Note that by Grothendieck vanishing, \(H^i(M) = 0\) for
\(i > {\left\lvert {\Phi^+} \right\rvert} = \dim(G/B)\), so this is a
finite sum. In fact, if \(M\) is a \(G{\hbox{-}}\)module, then this is
\(W{\hbox{-}}\)invariant and thus in fact
\(\chi(M) \in {\mathbb{Z}}[X(T)]^W\).

\hypertarget{wednesday-october-14}{%
\section{Wednesday, October 14}\label{wednesday-october-14}}

Today:

\begin{itemize}
\item
  Weyl's character formula
\item
  Strong linkage
\item
  Translation functors
\end{itemize}

Recall that we defined
\begin{align*}  
\operatorname{ch}(M) &\coloneqq\sum_{\mu \in X(T)} \qty{\dim M_\mu} e^{\mu} \in {\mathbb{Z}}[X(T)]\\
\chi(M) &\coloneqq\sum_{i\geq 0} (-1)^i \operatorname{ch}H^i(M) \in {\mathbb{Z}}[X(T)]^W
.\end{align*}

where \(H^i(M) = R^i \mathop{\mathrm{ind}}_B^G M\), and \(H^i(M) =0\)
for \(i> G/B = {\left\lvert {\Phi^+} \right\rvert}\).

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-14-14-03-12.png}
\caption{Image}
\end{figure}

Note that the Euler characteristic is additive on SESs: if
\(0\to A\to B\to C\to 0\) then \(\chi(B) = \chi(A) + \chi(B)\). It is
also multiplicative wrt the tensor product:
\(\chi(A\otimes B) \chi(A) \chi(B)\).

\todo[inline]{Because ?}

\begin{remark}

If \(\lambda \in X(T)_+\), then
\(\chi(\lambda) = \operatorname{ch}H^0(\lambda) = \operatorname{ch}(V(0))\).

\end{remark}

\begin{proposition}[?]

\hfill

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  The set
  \(\left\{{\operatorname{ch}L(\lambda) {~\mathrel{\Big|}~}\lambda\in X(T)_+}\right\}\)
  is a basis for \({\mathbb{Z}}[X(T)]^W\).
\item
  If \(\lambda \in X(T)\) and
  \(\sum a_\mu e^\mu \in {\mathbb{Z}}[X(T)]^W\), then there is a
  formula:
  \begin{align*}  
  \chi(\lambda) \qty{ \sum_\mu a_\mu e^\mu } = \sum_\mu a_\mu \chi(\lambda + \mu)
  .\end{align*}
\end{enumerate}

\end{proposition}

\begin{proof}[of 1]

Let
\begin{align*}
\operatorname{Sym}(\mu) \coloneqq\sum_{\nu \in W\mu} e^\nu
\end{align*}
be the sum over the \(W\) orbit of \(\mu\). This is clearly
\(W{\hbox{-}}\)invariant, so
\(\operatorname{Sym}(\mu) \in {\mathbb{Z}}[X(T)]^W\). Since every
\(\nu \in X(T)\) is \(W{\hbox{-}}\)conjugate to \(\mu\) (which is
dominant), the set
\(\left\{{\operatorname{Sym}(\mu) {~\mathrel{\Big|}~}\mu \in X(T)_+}\right\}\)
is a basis for \({\mathbb{Z}}[X(T)]^W\), since this set is linearly
independent.

\begin{quote}
Why: conjugate to a unique weight.
\end{quote}

Let \(\lambda \in X(T)_+\), then

\begin{align*}
\operatorname{ch}L(\lambda) = \operatorname{Sym}(\lambda) + \sum_{\substack{\mu < \lambda \\ \mu \in X(T)_+} } a_\mu \operatorname{Sym}(\mu)
.\end{align*}
Thus the transition matrix is unipotent and upper-triangular, thus
\(\left\{{\operatorname{ch}L(\lambda) {~\mathrel{\Big|}~}\lambda \in X(T)_+}\right\}\)
is a basis for \({\mathbb{Z}}[X(T)]^W\).

\end{proof}

\begin{proof}[of 2]

Since
\(\left\{{L(\lambda) {~\mathrel{\Big|}~}\lambda\in X(T)_+}\right\}\)
forms a basis for \({\mathbb{Z}}[X(T)]^W\), there is some
\(G{\hbox{-}}\)module \(V\) such that
\(\sum a_\mu e^\mu = \pm \operatorname{ch}V\). We can consider a
composition series of \(V\otimes\lambda\), where the factor
\(\left\{{\mu \otimes\lambda}\right\}\) appears \(a_\mu = \dim V_\mu\)
times. We now compute in two different ways:
\begin{align*}  
\chi(V\otimes\lambda) 
&= \operatorname{ch}(V) \chi(\lambda)  && \text{using the formula from earlier} \\
&= \chi(\lambda) \qty{ \sum_\mu a_\mu e^\mu }
.\end{align*}

On the other hand,
\begin{align*}  
\chi(V\otimes\lambda) &=
\sum a_\mu \chi(\lambda + \mu)
.\end{align*}

\end{proof}

\begin{remark}

The formula used above was
\begin{align*}  
R^i \mathop{\mathrm{ind}}_B^G (V\otimes\lambda) = V\otimes R^i \mathop{\mathrm{ind}}_B^G(\lambda)
.\end{align*}

\end{remark}

\hypertarget{weyls-character-formula-1}{%
\subsection{Weyl's Character Formula}\label{weyls-character-formula-1}}

For any \(\alpha\in\Delta\) and \(\lambda \in X(T)\) with
\({\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\).
We have an analog of Serre duality:
\begin{align*}  
\operatorname{ch}\mathop{\mathrm{ind}}_B^{P_\alpha} \lambda = \operatorname{ch}R^i \mathop{\mathrm{ind}}_B^{P_\alpha} s_\alpha \cdot \lambda
,\end{align*}
i.e.~the induced module coincides with the Weyl module.

By definition of the dot action, we have
\begin{align*}  
s_\alpha \cdot \lambda = s_\alpha(\lambda + \rho) - \rho
.\end{align*}

As in previous calculations, we have

\begin{align*}  
{\left\langle {s_\alpha\cdot\lambda},~{\alpha {}^{ \vee }} \right\rangle} = -{\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle} - 1 \leq - 1
.\end{align*}

As in the analysis of Bott-Borel-Weil, we have
\begin{align*}  
H^i(s_\alpha \cdot\lambda) &= H^i( R^1 \mathop{\mathrm{ind}}_B^{P_\alpha} s_\alpha\cdot\lambda ) \\
H^i(\lambda) &= H^i( \mathop{\mathrm{ind}}_B^{P_\alpha}\lambda )
,\end{align*}
since the spectral sequence collapses. Note that the two things
appearing on the RHS have the same Euler characteristics.

We can thus define define a modified Euler characteristic
\begin{align*}  
\phi(N) = \sum_{i\geq 0} (-1)^i \operatorname{ch}R^i \mathop{\mathrm{ind}}_{P_\alpha}^G(N)
.\end{align*}

and obtain \(\chi(\lambda) = -\chi(s_\alpha \cdot \lambda)\). The same
argument works for
\({\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} < 0\).

\begin{remark}[Very Important Fact]

\begin{align*}  
\lambda \in X(T) \implies \chi(\lambda) = -\chi(s_\alpha \cdot \lambda)
.\end{align*}

\end{remark}

\begin{proposition}[General Formula]

\begin{align*}  
\chi(w\cdot \lambda) = \operatorname{sgn}(w) \chi(\lambda) && \operatorname{sgn}(w) \coloneqq(-1)^{\ell(w)}
,\end{align*}
with the convention that \(\chi(0) = e^0 = 1\).

\end{proposition}

\begin{lemma}[?]

Let \(\lambda \in X(T)\) where
\(\sum a_\mu e^|mu \in {\mathbb{Z}}[X(T)]^W\), so (as we proved)
\begin{align*}  
\chi(\lambda) \qty{ \sum_\mu a_\mu e^\mu } = \sum_\mu a_\mu \chi(\lambda + \mu)
.\end{align*}
In the special case \(\lambda = 0\), we have
\(\chi(\lambda) = \chi(0) = e^0\), we obtain
\begin{align*}  
\sum_\mu a_\mu e^\mu = \sum_\mu a_\mu \chi(\mu)
.\end{align*}

Extend this to a field by letting
\(\lambda \in X(T) \otimes_{\mathbb{Z}}{\mathbb{Q}}\), then define
\begin{align*}  
A(\lambda) \coloneqq\sum_{w\in W} \operatorname{sgn}(w) e^{w \lambda} \in {\mathbb{Z}}[ (X(T) \otimes{\mathbb{Q}}]
.\end{align*}

Then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(w' A(\lambda) = \operatorname{sgn}(w') A(\lambda)\).
\item
  \(A(\mu) A(\lambda) \in {\mathbb{Z}}[X(T) \otimes{\mathbb{Q}}]^W\).
\end{enumerate}

\end{lemma}

Proof of 1: exercise.

\begin{proof}[of 2]

We can compute
\begin{align*}  
w(A(\mu) A(\lambda) ) 
&= w A(\mu)  w A(\lambda) \\
&= \operatorname{sgn}(w) A(\mu)  \operatorname{sgn}(w) A(\lambda) \\
&= \operatorname{sgn}(w)^2 A(\mu) A(\lambda) \\
&= A(\mu) A(\lambda)
.\end{align*}

\end{proof}

\begin{theorem}[Weyl's Character Formula]

Let \(\lambda \in X(T)\) be any weight, then
\begin{align*}  
\chi(\lambda) = { A(\lambda + \rho) \over A(\rho) }
,\end{align*}
where \(\rho = {1\over 2} \sum_{\alpha \in \Phi^+} \alpha\).

\end{theorem}

\begin{quote}
Note: this says that one formal sum divides another.
\end{quote}

A corollary is an analog of Weyl's dimension formula: :::\{.corollary
title=``?''\} Let \(\lambda \in X(T)_+\) be a dominant weight. Then
\begin{align*}  
\operatorname{ch}H^0(\lambda) = { A(\lambda + \rho) \over A(\rho) }
.\end{align*}
:::

Big question: suppose
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\).
What are \(\operatorname{ch}L(\lambda)\) and \(\lambda \in X(T)_+\)? We
know this for \(p\gg 0\), but in general it's wide open. There are
expressions in terms of ``\(p{\hbox{-}}\)bases'', but these are hard to
compute. There are only recursive formulas, none that are closed (and
these may not exist).

Next time:

\begin{itemize}
\tightlist
\item
  Proof of Weyl's character formula
\item
  Compute an example.
\end{itemize}

Idea of the proof: we'll have some
\(\chi(\lambda) = \sum_\mu a_\mu e^\mu\). Well also have
\(A(\rho) \qty{ \sum_\mu a_\mu e^\mu } = A(\lambda + \rho)\). This will
reduce to equating coefficients of two formal sums, which will result in
a system of linear equations.

\hypertarget{friday-october-16}{%
\section{Friday, October 16}\label{friday-october-16}}

\hypertarget{example-weyls-character-formula}{%
\subsection{Example: Weyl's Character
Formula}\label{example-weyls-character-formula}}

Review: suppose the following is invariant under the Weyl group, so
\(\sum a_\mu e^\mu \in {\mathbb{Z}}[X(T)]^W\). In this case, we have an
equality
\begin{align*}  
\sum a_\mu e^\mu = \sum a_\mu \chi(\mu)
,\end{align*}
where \(\chi(\mu) = \sum_{i\geq 0} (-1)^i \operatorname{ch}H^i(\mu)\).
We also had a relation
\begin{align*}  
\chi(w\cdot \mu) = (-1)^{\ell(w)} \chi(\mu) = \operatorname{sgn}(w) \chi(\mu)
.\end{align*}

Now let \(\lambda \in X(T) \otimes{\mathbb{Q}}\), then we defined
\begin{align*}  
A(\lambda) = \sum_{w\in W} \operatorname{sgn}(w) e^{w\lambda} \in {\mathbb{Z}}[X(T) \otimes{\mathbb{Q}}]
.\end{align*}

We obtain

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(w' A(\lambda) = \operatorname{sgn}(w') A(\lambda)\)
\item
  \(A(\mu) A(\lambda) = {\mathbb{Z}}[X(T) \otimes{\mathbb{Q}}]^W\).
\end{enumerate}

\begin{theorem}[Weyl's Character Formula]

\begin{align*}  
\lambda\in X(T) \implies \chi(\lambda) = {A(\lambda + \rho) \over A(\lambda)}
.\end{align*}

As a special case when \(\lambda \in X(T)_+\), all higher sheaf
cohomology vanishes and thus
\begin{align*}  
\operatorname{ch}H^0(\lambda) = {A(\lambda + \rho) \over A(\lambda)}
.\end{align*}

\end{theorem}

\begin{proof}

We first perform a \emph{reindexing} step:
\begin{align*}  
\sum_{w, w'} \operatorname{sgn}(w\cdot w') e^{w(\lambda+\rho) + w'\rho}
&= \sum_{w, w'} \operatorname{sgn}(w^{-1} w') e^{w(\lambda+\rho) + w'\rho} \\
&= \sum_{w, y} \operatorname{sgn}(y) e^{w(\lambda+\rho) + wy\rho} && y = w^{-1}w' \implies w' = wy \\
&= \sum_{w, y} \operatorname{sgn}(y) e^{w(\lambda + \rho + y\rho)}
.\end{align*}

Now let \(\lambda\in X(T)\), we then compute
\begin{align*}  
A(\lambda + \rho) A(\rho)
&=
\sum_{w} \operatorname{sgn}(w) e^{w(\lambda + \rho)}
+ \sum_{w'} \operatorname{sgn}(w') e^{w'(\lambda + \rho)}  \\
&=
\sum_{w, w'} \operatorname{sgn}(ww') e^{w(\lambda + \rho) + w'\rho} \\
&= 
\sum_{w, w'} \operatorname{sgn}(w') e^{w(\lambda + \rho + w'\rho)} && \text{from reindexing above, setting } y\coloneqq w' \\
&= \sum_{w, w'} \operatorname{sgn}(w') \chi\qty{w(\lambda + \rho + w'\rho)} \\
&= \sum_{w, w'} \operatorname{sgn}(w') \chi\qty{w\cdot (\lambda + w'\rho + w^{-1} \rho)} && \text{definition of dot action}\\
&= \sum_{w, w'} \operatorname{sgn}(ww') \chi\qty{\lambda + w'\rho + w\rho }  && \text{swapping } w\leadsto w^{-1}  
.\end{align*}

Note that \(\chi\) can be introduced since
\(A(\lambda + \rho)A(\rho) \in {\mathbb{Z}}[X(T) \otimes{\mathbb{Q}}]^{W\cdot}\).

\todo[inline]{Not sure, double check.}

We can now conclude that
\begin{align*}  
A(\rho)^2 = \sum_{w, w'} \operatorname{sgn}(ww') e^{w\rho + w' \rho}
.\end{align*}
Since this quantity is \(W{\hbox{-}}\)invariant, since it's a square, we
can move the \(\chi\) inside:
\begin{align*}  
\chi(\lambda) \qty{ \sum a_\mu e^\mu } = \sum a_\mu \chi(\lambda + \mu) \\
\implies \chi(\lambda) A(\rho)^2 = \sum_{w, w'} \operatorname{sgn}(ww') \chi(\lambda + w\rho + w'\rho)
,\end{align*}
which is exactly what the first calculation resulted in. So we can
conclude
\begin{align*}  
A(\lambda + \rho) A(\rho) = \chi(\lambda) A(\rho)^2
.\end{align*}
Note that \(A(\rho) \neq 0\) since \(w\rho \neq \rho\) unless
\(w=\operatorname{id}\). Thus we are actually working in
\({\mathbb{Z}}[X(T) + {\mathbb{Z}}\rho]\), which is an integral domain,
and thus we can apply cancellation laws to obtains
\begin{align*}  
A(\lambda + \rho) = \chi(\lambda) A(\rho)
.\end{align*}

\end{proof}

\begin{example}

Let \(G = \operatorname{GL}_3(k)\), which has a natural 3-dimensional
representation \(V\). Let \(\lambda = (1,0,0)\), so \(L(1,0,0) = V\).
This is a polynomial representation, so by permuting we can obtain
\begin{align*}  
\operatorname{ch}V = e^{(1,0,0)} + e^{(0,1,0)} + e^{(0,0,1)} = \chi(1,0,0)
,\end{align*}
where the last equality holds since \(\lambda\) is dominant.

We can write \(\rho = (2,1,0)\), since the fundamental weights are given
by \(w_1 = (1,0,0)\) and \(w_2 = (1,1,0)\) (since we're in an
\({\operatorname{SL}}_2\) and/or \(A_2\) situation). We then obtain
\(\lambda + \rho = (3,1,0)\), and since \(W= S_3\),
\begin{align*}  
A(\lambda + \rho) = \sum_{w\in W} \operatorname{sgn}(w) e^{w(\lambda + \rho)}
=
e^{(3,1,0)} -
e^{(1,3,0)} + 
e^{(1,0,3)} - 
e^{(0,1,3)} + 
e^{(0,3,1)} - 
e^{(3,0,1)}
.\end{align*}

Thus
\begin{align*}  
A(\rho) =
e^{(2,1,0)} -
e^{(1,2,0)} + 
e^{(1,0,2)} - 
e^{(0,1,2)} + 
e^{(0,2,1)} - 
e^{(2,0,1)}
.\end{align*}

We can then compute
\begin{align*}  
\chi(1,0,0) A(\rho) = &e^{(3,1,0)}
- e^{(2,2,0)} + 
e^{(2,0,2)} 
-e^{(1,1,2)} + 
e^{(1,2,1)}
- e^{(3,0,1)} + 
\\
&e^{(2,2,0)} -
e^{(1,3,0)} + 
e^{(1,1,2)} - 
e^{(0,2,2)} + 
e^{(0,3,1)} - 
e^{(2,1,1)} + 
\\
&e^{(2,1,1)} -
e^{(1,2,1)} + 
e^{(1,0,3)} - 
e^{(0,1,3)} + 
e^{(0,2,2)} - 
e^{(2,0,2)}
.\end{align*}

After cancellation, you'll find that this expression is equal to
\(A(\lambda + \rho)\).

\end{example}

\hypertarget{strong-linkage-principle}{%
\subsection{Strong Linkage Principle}\label{strong-linkage-principle}}

We'll consider representations in characteristic zero, so we can take
\(k={\mathbb{C}}\). Let \(G\) bet a complex simple group,
\({\mathfrak{g}}= \operatorname{Lie}(G)\), \(t\) a maximal torus, \(X\)
the weights, and \(X_+\) the dominant weights. We have a correspondence
\Large
\begin{align*}  
\left\{{\substack{(g, t)}}\right\} \iff
\left\{{\substack{(\Phi, W)}}\right\}
\end{align*}
\normalsize

where \(\Phi\) is an irreducible root system and \(W\) is the Weyl
group. We'll have a set of simple roots \(\Delta\subseteq \Phi^+\). For
\(\lambda\in X\), we have
\begin{align*}  
Z(\lambda) = U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}}^+)} \lambda \twoheadrightarrow L(\lambda)
.\end{align*}

Then \(\lambda \in X_+ \iff L(\lambda)\) is finite dimensional. We have
\(W\) acting on \(X\) via reflections, which we can extend to a dot
action
\begin{align*}  
w\cdot \lambda = w(\lambda + \rho) - \rho, \hspace{4em} \rho = {1\over 2}\sum_{\alpha\in\Phi^+} \alpha
.\end{align*}

We define Category \({\mathcal{O}}\) which has objects
\({\mathfrak{g}}{\hbox{-}}\)modules with a weight space decomposition
which is locally finite wrt \({\mathfrak{n}}^+\).

\hypertarget{linkage-in-category-mathcalo}{%
\subsubsection{\texorpdfstring{Linkage in Category
\({\mathcal{O}}\)}{Linkage in Category \{\textbackslash mathcal\{O\}\}}}\label{linkage-in-category-mathcalo}}

Set \(Z(\lambda) = \Delta(\lambda)\), then
\begin{align*}  
[Z(\lambda) : L(\mu)] \neq 0 \implies \lambda \in W\cdot \mu
.\end{align*}
The LHS is computed by evaluating certain Kazhdan-Lusztig polynomials at
\(x=1\).

\begin{example}

Let \(\Phi= A_2\), then

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-16-14-43-26.png}
\caption{Image}
\end{figure}

\({\mathcal{O}}_0\) is the principal block, and the irreducibles
correspond to
\(\left\{{L(w\cdot 0) {~\mathrel{\Big|}~}w\in W}\right\}\), and the
number of irreducibles in given by \({\left\lvert {W} \right\rvert}\).
In this case, there is only 1 finite-dimensional module in any given
block of category \({\mathcal{O}}\).

\end{example}

\begin{example}

For \(\Phi = A_1\), we have the following situation:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-16-14-46-27.png}
\caption{Image}
\end{figure}

In \({\mathcal{O}}_0\), there are two irreducible representations given
by the Verma modules \(L(0), L(-2)\), and we find that

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-16-14-48-12.png}
\caption{Image}
\end{figure}

In this case, the projectives are given by

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-16-14-51-51.png}
\caption{Image}
\end{figure}

\end{example}

\hypertarget{monday-october-19}{%
\section{Monday, October 19}\label{monday-october-19}}

\todo[inline]{Missing notes from first 10m! See phone screenshot.}

\hypertarget{representations-in-positive-characteristic}{%
\subsection{Representations in Positive
Characteristic}\label{representations-in-positive-characteristic}}

We have the following setup:
\begin{align*}  
G && \text{a semisimple, simply connected algebraic group} \\
k && \text{an algebraically closed field of characteristic $p>0$} \\
T && \text{a maximal torus} \\
B && \text{a Borel (negative roots)} \\
X(T) = X && \text{weights} \\
X(T)_+ = X_+ && \text{dominant weights} \\
\Phi && \text{roots}
.\end{align*}

For \(\lambda \in X_+\), we consider the induced module
\(H^0(\lambda) = \mathop{\mathrm{ind}}_B^G \lambda\). Not that this is
not a simple module in general, so we instead ask about its composition
factors.

Question: For all \(\lambda, \mu \in X_+\), what are the multiplicities
\([H^0(\lambda): L(\mu)]\).

\begin{example}

Let \(G = {\operatorname{SL}}_2(k)\), so \(\Phi = A_1\). Then
\(\lambda \in X_+ = \left\{{0,1,2,\cdots}\right\}\) as we know from
standard facts in lie algebras. Define
\(X_1 = \left\{{0, 1, \cdots, p-1}\right\}\), then
\(\dim H^0(\lambda) = \lambda + 1\). We can write the weight
\(p{\hbox{-}}\)adically as \(\lambda = \sum_{i=0}^t \lambda_i p^i\) for
some \(\lambda_j\in X_1\). Thus
\(L(\lambda) = L(\lambda_0) \bigotimes_{i=1}^t L(\lambda_i)^{(i)}\).

Consider \(p=3, \lambda = 7\), then \(\dim H^0(7) = 8\). We can write
\(7\) 3-adically as \(7 = (1)3^0 + (2)3^1\), and so
\begin{align*}  
L(7) \cong L(1) \otimes L(2)^{(1)}
.\end{align*}
The first summand is 2-dimensional, and the second is 3-dimensional, so
\(L(7)\) is 6-dimensional. Note that \(L(7) \hookrightarrow H^0(7)\).

We can calculate the weights in the tensor product: the first has
weights \(\left\{{\pm 1}\right\}\), we take the adjoint weights in the
second factor and multiply by the twist 3 to get
\(\left\{{2\cdot 3, 0\cdot 3, -2\cdot 3}\right\}\). Taking all
combinations of sums from these yields
\(\left\{{7,5,1,-1,-5,-7}\right\}\).

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-19-14-14-46.png}
\caption{Comparing what's left over}
\end{figure}

Since \(\pm 3\) are left over, we know \([H^0(7): L(3)] \neq 0\). We can
continue with \(3 = (1)3^1\) and write \(L(3) = L(1)^{(1)}\). We get
weights of the form \(1\cdot 3, 1\cdot -3\), so nothing is left over and
we're done. We thus get a decomposition

\begin{center}
\begin{tikzcd}
 & & L(3) \ar[dd] \\
H^0(7): & & \\
 & & L(7)
\end{tikzcd}
\end{center}

Note the difference to Verma modules in category \({\mathcal{O}}\): we
have to consider the action of the \emph{affine} Weyl group, where
\(W_a \coloneqq W \rtimes p{\mathbb{Z}}\Phi\). Here we have hyperplanes
at \(p-1, 2p-1, 3p-1\), and 7 is \emph{linked} to 3 (in the same orbit)
for this action:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-19-14-21-35.png}
\caption{Image}
\end{figure}

\end{example}

\begin{quote}
Once characters are known, can find composition factors.
\end{quote}

\hypertarget{affine-weyl-group}{%
\subsection{Affine Weyl Group}\label{affine-weyl-group}}

Letting \(a\in {\mathbb{N}}\), we have
\(W_a = W\rtimes a({\mathbb{Z}}\Phi)\) where \({\mathbb{Z}}\Phi\) is the
root lattice. Note that there are other variants:

\begin{itemize}
\tightlist
\item
  \(W_a = W\rtimes a({\mathbb{Z}}\Phi {}^{ \vee })\),
\item
  \(W_{\text{ext}} = W \rtimes X(T)\).
\end{itemize}

So we set \(W_p = W\rtimes p({\mathbb{Z}}\Phi)\) where \(p\) is a prime.
What's in this group? We know it contains ``products'' of reflections
with translations. We find that \(W_p\) is generated by
\begin{align*}  
s_{\beta, np}(\lambda) = \lambda - {\left\langle {\lambda},~{\beta {}^{ \vee }} \right\rangle}\beta + np \beta
.\end{align*}

It is also the case that \(W_p\) acts on \(X(T)\) and there exists a dot
action
\begin{align*}  
w\cdot \lambda = w(\lambda + \rho) - \rho
.\end{align*}

\begin{example}

Consider \(A_1\), so \(\alpha = 2\). We consider what the stabilizer is:
\begin{align*}  
s_{\alpha, np}\cdot \lambda &= \lambda \\
s_{\alpha, np}(\lambda + \rho) - \rho &= \lambda \\
(\lambda + \rho) - {\left\langle {\lambda _ \rho},~{\alpha {}^{ \vee }} \right\rangle}\alpha + np\alpha - \rho &= \lambda
.\end{align*}

After cancellation in the last line above, we obtain
\begin{align*}  
\lambda = np-1
,\end{align*}
which exactly yields the \(p-1, 2p-1, \cdots\) we saw before.

\end{example}

\begin{example}

Consider \(A_2\). We obtain ``alcoves'':

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-19-14-36-02.png}
\caption{Image}
\end{figure}

\end{example}

We can get a stronger version of weak linkage, which we'll just call
linkage:

\begin{theorem}[Linkage]

\begin{align*}  
[H^0(\lambda): L(\mu)] \neq 0 \implies \lambda \in W_p \cdot \mu
.\end{align*}

\end{theorem}

\begin{warnings}

These are difficult to compute in general, or to even detect when
they're zero. For \(p\gg 0\), these multiplicities are computed via
Kazhdan-Lusztig polynomials.

\end{warnings}

\hypertarget{ordering-of-weights}{%
\subsubsection{Ordering of Weights}\label{ordering-of-weights}}

There is a partial ordering on the weight lattice given by
\begin{align*}  
\mu \leq \lambda \iff \lambda - \mu = \sum_{\alpha\in \Phi^+} n_\alpha \alpha, \quad n_\alpha \geq 0
.\end{align*}

\begin{definition}[Strong Linkage]

For \(\mu, \lambda \in X(T)\), we say \(\mu\) is \textbf{strongly
linked} to \(\lambda\), denoted \(\mu \uparrow \lambda\), if there
exists a sequence of weights \(\mu_1, \cdots, \mu_r \in X(T)\) and
reflections \(s_1, \cdots, s_r\) such that
\begin{align*}  
\mu \leq \mu_1 = s_1 \cdot \mu \leq \mu_2 = s_2\cdot \mu 1 \leq \cdots \leq s_r \mu_{r-1}
.\end{align*}

\end{definition}

\begin{remark}

Note that

\begin{itemize}
\tightlist
\item
  \(\mu \uparrow \lambda \implies \mu \leq \lambda\), so this is
  stronger than the usual linkage
\item
  \(\mu \uparrow \lambda \implies \mu \in W_p \cdot \lambda\).
\end{itemize}

\end{remark}

\begin{theorem}[Strong Linkage Principle]

\begin{align*}  
[H^0(\lambda): L(\mu)] \neq 0 \implies \lambda \in \mu \uparrow \lambda
.\end{align*}

Moreover, there is a version of strong linkage for \(H^i(\lambda)\) for
\(i> 1\).

\end{theorem}

\begin{quote}
Next time: history of strong linkage, and translation functors.
\end{quote}

\hypertarget{wednesday-october-21}{%
\section{Wednesday, October 21}\label{wednesday-october-21}}

\hypertarget{strong-linkage}{%
\subsection{Strong Linkage}\label{strong-linkage}}

Let \(G\) be a semisimple algebraic group and
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\).
We found that the \emph{affine Weyl group} \(W_p\) played an important
role here.

\begin{theorem}[Strong Linkage I]

Suppose we have a nonzero composition factor in the induced/Weyl module.
Then
\begin{align*}
[H^0 \lambda : L(\mu)] \neq 0]\implies \mu \uparrow \lambda
.\end{align*}

In other words, there's a series of reflections sending \(\mu\) to
\(\lambda\) which doesn't increase it's value in the ordering.

\end{theorem}

\begin{theorem}[Strong Linkage II]

Let \(\lambda \in X(T)\) with
\({\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \geq 0\)
for all \(\alpha\in \Delta\). Suppose \(\mu \in X(T)_+\).
\begin{align*}
[H^i w\cdot \lambda : L(\mu)] \neq 0 \text{ for some } i\geq 0 \implies \mu \uparrow \lambda
.\end{align*}

\end{theorem}

\begin{remark}

Note that this is tells us slightly more than Bott-Borel-Weil.

\end{remark}

\begin{remark}

There is some history here:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Verma conjectured the first theorem in 1971.
\item
  Humphreys (1971) proved it for
  \(Z_r(\lambda) = \mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda\).
\item
  Strong Linkage II proved by Andersen in 1980.
\item
  Jantzen proved strong linkage for \(Z_r\), which implies strong
  linkage for \(V(\lambda)\).
\item
  Doty (1987) proved strong linkage for \(Z_r(\lambda)\) as a
  \(G_rT{\hbox{-}}\)modules, which implies strong linkage for
  \(V(\lambda)\).
\end{enumerate}

\end{remark}

\begin{remark}

One application is the following: let \(\lambda, \mu \in X(T)_+\), then
\(\operatorname{Ext} _G^n(L(\lambda), L(\mu)) \neq 0\) for some
\(n \geq 0\). This implies that \(\lambda \in W_p \cdot \mu\).

We can consider some cases

\begin{itemize}
\tightlist
\item
  If \(n=0\), we're reduced to previous situations.
\item
  If \(n=1\), we can conclude that \(L(\lambda)\) is in the second socle
  layer of \(H^0 \mu\), or vice-versa. In either case,
  \(\lambda \in W_p \cdot \mu\).
\end{itemize}

We can compute this ext by considering an minimal injective resolution
\begin{align*}  
0 \to L(\mu) \to I_0 = I(\mu) \to I_1 \to \cdots
.\end{align*}

We can conclude that
\begin{align*}
[I(\mu) : H^0(\sigma)] = [H^0(\sigma): L(\mu)] \neq 0
.\end{align*}
by Brauer-Humphreys reciprocity, so \(\sigma \in W_p \cdot \mu\).
Similarly \([I(\mu): L(\gamma)] \neq 0\) implies that
\(\gamma \in W_p \cdot \mu\), and continuing in this way we can write
\begin{align*}
I_1 = \bigoplus_{j=1}^t I(\gamma_j) 
\text{ with each } 
\gamma_j \in W_p \cdot \mu
.\end{align*}
So all of these weights are strongly linked to \(\mu\).

But then we know \(\operatorname{Ext} _G^n (L(\lambda), L(\mu)) \neq 0\)
is a subquotient of \(\hom_G(L(\lambda), I_n)\), which thus can not be
zero. So \(\lambda \in W_p \cdot \mu\)

\end{remark}

\hypertarget{translation-functors}{%
\subsection{Translation Functors}\label{translation-functors}}

Consider the case from category \({\mathcal{O}}\), e.g.~by taking
\({\mathfrak{g}}= {\mathfrak{sl}}_3({\mathbb{C}})\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-21-14-19-53.png}
\caption{Image}
\end{figure}

For \(\lambda\) a regular weight, the principal block \(\mathcal{B}_0\)
is Morita-equivalent to \(\mathcal{B}_\lambda\). If \(\mu\) is a
singular weight, then by Jantzen there are translation functors

\begin{align*}  
T_\lambda^\mu: \mathcal{B}_\lambda &\to \mathcal{B}_\mu \\
T_\mu^\lambda: \mathcal{B}_\mu &\to \mathcal{B}_\lambda
.\end{align*}

In the case where \(G\) is a semisimple algebraic group and
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}\mkern-1.5mu}\mkern 1.5mu_p\),
we have the following picture instead:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-21-14-23-24.png}
\caption{Image}
\end{figure}

\hypertarget{blocks}{%
\subsubsection{Blocks}\label{blocks}}

Two simple modules \(S, T\) are in the same \emph{block} if we have a
sequence \(T_1, \cdots, T_n\) such that \(S=T_1\) and \(T_n = T\) where
\(\operatorname{Ext} ^1(T_i, T_{i+1}) \neq 0\).

\begin{lemma}[?]

Let \(M, M'\) be \(H{\hbox{-}}\)modules and \(\mathcal{B}(H)\) be the
blocks of \(H\). Then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(M = \bigoplus_{b\in \mathcal{B}(H)} M_b\) where
  \(M_b = \sum_{M'\leq M} M'\) the sum of all submodules such that \(M\)
  has composition in the block \(b\).
\item

  \begin{align*} \operatorname{Ext} _H^i(, M') = \prod_{b\in\mathcal{B}(H)} \operatorname{Ext} _H^i (M_b, M_b') \end{align*}
\end{enumerate}

\end{lemma}

So the question becomes, what are the blocks of \(H\)? Let
\(\lambda \in X(T)_+\), so we can define \(L(\lambda)\), and let
\(b(\lambda)\) be the \(G{\hbox{-}}\)block containing \(L(\lambda)\).

We have \(b(\lambda) \in \mathcal{B}(G)\) and \(b(\lambda)\)
\(\subseteq X(T)_+ \cap W_p \cdot \lambda\), i.e.~we have strong
linkage.

\begin{quote}
Here we refer to \(b(\lambda)\) as both the block and the weights it
contains.
\end{quote}

\begin{theorem}[Donkin]

Let \(\lambda \in X(T)_+\) be a dominant weight and let
\(r\in {\mathbb{Z}}\) be the largest integer such that
\(p^r \divides {\left\langle { \lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle}\)
for all \(\alpha\in \Phi\). Then
\begin{align*}
b(\lambda) = W_p^{(r)} \cdot \lambda \cap X(T)_+ \text{ where } W_p^{(r)} = W\rtimes p^r {\mathbb{Z}}\Phi
.\end{align*}

\end{theorem}

\begin{proposition}[?]

Let \(B\) be a \(G{\hbox{-}}\)module and \(\lambda \in X(T)\). Set
\({\operatorname{pr}}_\lambda V\) to be the sum of all submodules of
\(V\) with composition factors of the form \(L(\mu)\) where
\(\mu \in W_p \cdot \lambda\). Then

\begin{itemize}
\item
  \(V = \bigoplus_{\lambda \in Z} {\operatorname{pr}}_\lambda V\) where
  \(Z\) are representatives of the \(W_p\) orbits, i.e.~one
  representative from each alcove in the weight lattice.
\item

  \begin{align*} \operatorname{Ext} _G^i(V, V') = \prod_{\lambda \in Z} \operatorname{Ext} _G^i ({\operatorname{pr}}_\lambda V, {\operatorname{pr}}_\lambda V') \end{align*}
\item
  The projection functors \({\operatorname{pr}}_\lambda({-})\) are
  exact.
\end{itemize}

\begin{quote}
Note that this still works for singular weights, not just regular
weights.
\end{quote}

\end{proposition}

\begin{example}

We can compute
\begin{align*}  
{\operatorname{pr}}_\lambda L(\mu) = 
\begin{cases}
0 &= \lambda \not\in W_p \cdot \mu \\
L(\mu) &= \lambda \in W_p \cdot \mu
\end{cases}
.\end{align*}

Similarly, by strong linkage,
\begin{align*}  
{\operatorname{pr}}_\lambda H^i(\mu) =
\begin{cases}
0 &= \lambda \not\in W_p \cdot \mu \\
H^i(\mu) &= \lambda \in W_p \cdot \mu
\end{cases}
.\end{align*}

\end{example}

Recall that
\begin{align*}  
\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\coloneqq\left\{{
\lambda \in X(T) {~\mathrel{\Big|}~}
0 \leq {\left\langle {\lambda + \rho},~{\beta {}^{ \vee }} \right\rangle} \leq p \,\, \forall \beta\in\Phi^+
}\right\}
.\end{align*}
For every
\(\mu, \lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\),
consider \(\mu - \lambda \in X(T)\). Then there is a way to conjugate it
under the ordinary \(W\) action to land in the dominant region,
i.e.~some unique \(\nu\) such that
\(\nu \in X(T)_+ \cap W(\mu - \lambda)\).

\begin{definition}[Translation Functors]

Define
\begin{align*}  
T_\lambda^\mu V = 
{\operatorname{pr}}_\mu
\qty{
L(\nu) \otimes
{\operatorname{pr}}_\lambda V
}
.\end{align*}

So project to \(\lambda\), tensor with an irreducible representation,
then project to \(\mu\). This is an exact functor
\begin{align*}  
T_{\lambda}^\mu: G{\hbox{-}}\mathrm{mod} &\to G{\hbox{-}}\mathrm{mod}
.\end{align*}

\end{definition}

Next time: we'll show that \(T_\lambda^\mu\) and \(T_\mu^\lambda\) form
an adjoint pair. Note that if \(\mu, \lambda\) are in the same block,
these are the exact functor which product the categorical equivalence.

\hypertarget{friday-october-23}{%
\section{Friday, October 23}\label{friday-october-23}}

\hypertarget{facets}{%
\subsection{Facets}\label{facets}}

\(W_p\) has a dot action on
\(E \coloneqq X(T) \otimes_{\mathbb{Z}}{\mathbb{R}}\).

\begin{definition}[Facet]

We can write \(\Phi^+ = \Phi_0^+ \cup\Phi_1^+\), and define the
\emph{facet} as
\begin{align*}  
F \coloneqq\left\{{
\lambda \in E {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = n_\alpha p\,\, \forall\alpha\in \Phi_0^+(F),\,\,
(n_\alpha - 1)p < {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} < n_\alpha p \,\,\forall \alpha\in \Phi_1^+(F)
}\right\}
.\end{align*}
The first condition corresponds to being on a vertex in the following
diagram, while the second corresponds to being in the interior of a
triangle:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-23-14-04-42.png}
\caption{Image}
\end{figure}

\end{definition}

\begin{definition}[Closure of a Facet]

The \emph{closure} of a facet is defined by replacing the second
condition with an inequality.
\begin{align*}  
\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu \coloneqq\left\{{
\lambda \in E {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = n_\alpha p\,\, 
\forall\alpha\in \Phi_0^+(F),\,\,
n_\alpha - 1 \leq  {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \leq n_\alpha p 
\,\,\forall \alpha\in \Phi_1^+(F)
}\right\}
.\end{align*}
This includes all of the walls of the triangle.

\end{definition}

\begin{definition}[Upper Closure of a Facet]

Finally, we define the \emph{upper closure} by replacing one inequality
with a strict inequality:
\begin{align*}  
\widehat{F} \coloneqq\left\{{
\lambda \in E {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = n_\alpha p\,\, \forall\alpha\in \Phi_0^+(F),\,\,
n_\alpha - 1 < {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \leq n_\alpha p \,\,\forall \alpha\in \Phi_1^+(F)
}\right\}
.\end{align*}

\end{definition}

\begin{definition}[Alcove]

A facet is called an \textbf{alcove} for \(W_p\) iff
\(\Phi_0^+(F) = \emptyset\).

\end{definition}

\begin{remark}

Note that if \(F\) is an alcove for \(W_p\), then \(\widehat{F}\) is a
fundamental domain for \(W_p\curvearrowright E\) with the dot action.

\end{remark}

\hypertarget{translation-functors-1}{%
\subsection{Translation Functors}\label{translation-functors-1}}

Let
\(\lambda, \mu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\),
and define
\begin{align*}  
T_\lambda^\mu({-}) \coloneqq
{\operatorname{pr}}_\mu \qty{ L(\nu_1) \otimes{\operatorname{pr}}_\lambda({-}) }\\
\text{where } 
\nu_1 \in X(T)_+ \cap W(\mu-\lambda)
.\end{align*}

This is exact as a composition of exact functors, since we're tensoring
over a field and taking projections (which are themselves exact).

\begin{lemma}[?]

Let \(\lambda,\mu\in X(T)\) and \(M\) be a finite-dimensional
\(G{\hbox{-}}\)module. Then the functors
\begin{align*}  
F({-}) &\coloneqq{\operatorname{pr}}_\mu \circ \qty{M\otimes_k {-}} \circ {\operatorname{pr}}_\lambda \\
G({-}) &\coloneqq{\operatorname{pr}}_\lambda \circ \qty{M {}^{ \vee }\otimes_k {-}} \circ {\operatorname{pr}}_\mu \\
\end{align*}
define an adjoint pair, i.e.
\begin{align*}  
\hom_\mathcal{C}(G({-}), A) &= \hom_\mathcal{D}({-}, F(A)) \\
\hom_\mathcal{C}({-}, G(A)) &= \hom_\mathcal{D}( F({-}), {-}) \\
.\end{align*}

\end{lemma}

\begin{proof}

Let \(V, V'\) be \(G{\hbox{-}}\)modules. Then
\begin{align*}  
\hom_G(FV, V') 
&= \hom_G( {\operatorname{pr}}_\mu\qty{ M\otimes{\operatorname{pr}}_\lambda V  }, V') \\
&= \hom_G(  M\otimes{\operatorname{pr}}_\lambda V  , {\operatorname{pr}}_\mu V') \\
&= \hom_G(  {\operatorname{pr}}_\mu \qty{ M\otimes{\operatorname{pr}}_\lambda V }  , {\operatorname{pr}}_\mu V') \\
&= \hom_G(  {\operatorname{pr}}_\lambda V  , M {}^{ \vee }\otimes_k {\operatorname{pr}}_\mu V') \\
&= \hom_G(  {\operatorname{pr}}_\lambda V  , {\operatorname{pr}}_\lambda\qty{ M {}^{ \vee }\otimes_k {\operatorname{pr}}_\mu V'} ) \\
&= \hom_G(  V  , {\operatorname{pr}}_\lambda\qty{ M {}^{ \vee }\otimes_k {\operatorname{pr}}_\mu V'} ) \\
&= \hom_G(V, GV')
.\end{align*}

Here we've used the fact that there no nontrivial homs between distinct
blocks.

\end{proof}

\begin{theorem}[?]

Let
\(\lambda, \mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
are in the closure of the bottom alcove. Then
\(T_\lambda^\mu \leftrightarrows T_\mu \lambda\) form an adjoint pair.

\end{theorem}

\begin{proof}

Applying the previous corollary, we just need to show the last equality
in the following:
\begin{align*}  
T_\lambda^\mu({-}) 
&= {\operatorname{pr}}_\mu \qty{ L(\nu_1) \otimes{\operatorname{pr}}_\lambda({-}) } \\
&= {\operatorname{pr}}_\lambda \qty{ L(\nu_1) {}^{ \vee }\otimes{\operatorname{pr}}_\mu({-}) } \\
&=_? T_\mu^\lambda
.\end{align*}

This requires checking the highest weight condition on
\(L(\nu_1) {}^{ \vee }= L(-w_0 \nu_1)\). We know
\(\nu_1 \in X(T)_+ \cap W(\mu-\lambda)\), so if
\(\nu_1 = w(\mu-\lambda)\), we have
\(-w_0 \nu_1 = w_0 w (\lambda - \mu) \in W(\lambda - \mu)\). Since
\(-w_0 \nu_1 \in X(T)_+\), this verifies the condition.

\end{proof}

\begin{remark}

The adjointness can be extended from homs to exts:
\begin{align*}  
\operatorname{Ext} _G^i(T_\mu^\lambda V, V' ) \cong \operatorname{Ext} _G^i(V, T_\lambda^\mu V')
.\end{align*}

\end{remark}

\hypertarget{technical-preliminaries}{%
\subsection{Technical Preliminaries}\label{technical-preliminaries}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  If \(\lambda \in X(T)\) and
  \begin{align*}  
    \sum_\mu a(\mu) e^\mu \in {\mathbb{Z}}[X(T)]^W
    \end{align*}
  is \(W{\hbox{-}}\)invariant, then we proved that
  \begin{align*}  
    \chi(\lambda)
    \qty{
    \sum_\mu a(\mu) e^{\mu}
    }
    = \sum_\mu a(\mu) \chi(\lambda + \mu)
    .\end{align*}
\item
  If \({\operatorname{pr}}_\lambda V = V\), then we have
  \begin{align*}  
  \operatorname{ch}(M\otimes V) 
  &= \operatorname{ch}(M) \operatorname{ch}(V) \\
  &= \operatorname{ch}(M) \qty{\sum_{w\in W_p} a_w \chi(w\cdot\lambda) } \\
  &= \qty{ \sum_{\nu \in X(T)} \dim M_\nu e^\nu } \qty{\sum_{w\in W_p} a_w \chi(w\cdot\lambda) }
  .\end{align*}
\end{enumerate}

\begin{proposition}[?]

Let \(V\) be a finite dimensional \(G{\hbox{-}}\)module with
\({\operatorname{pr}}_\lambda V = V\). Write
\begin{align*}  
\operatorname{ch}(V) = \sum_{w\in W_p} a_w \chi(w\cdot \lambda) \quad
a_w\in {\mathbb{Z}},\, \text{cofinitely zero}
.\end{align*}
Then
\begin{align*}  
\operatorname{ch}\qty{{\operatorname{pr}}_\lambda \qty{ M\otimes V } } = 
\sum_{w\in W} a_w  \qty{ \sum_{\substack{ \nu \in X(T) \\ \lambda + \nu \in W_p\cdot \mu} } \dim M_\nu } \chi(w\cdot (\lambda + \nu) )
.\end{align*}

\end{proposition}

\begin{proof}

Using (1) and (2), we can write
\begin{align*}  
\operatorname{ch}(M\otimes V) = 
\sum_{w\in W_p} a_w \sum_\nu \dim M_\nu \chi(w\cdot \lambda + \nu)
.\end{align*}
Note that \(w\cdot \lambda + \nu = w\cdot (\lambda + w_1 \nu)\) where
\(w_1 \coloneqq w^{-1}\), using the fact that the dot action acts
linearly on the second term. This comes from the following computation:
\begin{align*}  
w\cdot(\mu_1 + \mu_2)
&= w(\mu_1 + \mu_2 - \rho) + \rho \\
&= w(\mu_1 + \rho) - \rho + w\mu_2 \\
&= w\cdot \mu_1 + w\mu_2
.\end{align*}

We can thus write
\begin{align*}  
\operatorname{ch}(M\otimes V) = 
\sum_{w\in W_p} a_w \qty{ \sum_\nu \dim M_\nu \chi(w\cdot \qty{\lambda + \nu})}
,\end{align*}
since summing over \(\nu\) is the same as summing over \(w\nu\) for any
\(w\).

To get \(\operatorname{ch}({\operatorname{pr}}_\mu(M\otimes V))\), take
\(\chi(w(\lambda + \nu))\) and note that
\(\lambda + \nu \in W_p \cdot \mu\).

\end{proof}

\begin{remark}

Given \(\operatorname{ch}V\), one can write
\(\operatorname{ch}T_\lambda^\mu V\). What will be important here are
stabilizers. If \(\lambda\) is on a wall, the stabilizer fixes the
corresponding hyperplane.

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-23-14-50-27.png}
\caption{Image}
\end{figure}

\end{remark}

\hypertarget{monday-october-26}{%
\section{Monday, October 26}\label{monday-october-26}}

\hypertarget{review-2}{%
\subsection{Review}\label{review-2}}

Let \(V\) be a finite dimensional \(G{\hbox{-}}\)module with
\({\operatorname{pr}}_\lambda V = V\), and write
\(\operatorname{ch}(V) = \sum_{w\in W_p} a_w \chi(w\cdot \lambda)\)
where \(a_w\in {\mathbb{Z}}\) and only finitely many are nonzero. We can
then write
\begin{align*}  
\operatorname{ch}\qty{{\operatorname{pr}}_\mu(M\otimes V)} = \sum_{w\in W_p} a_w \qty{\sum_{\substack{\nu \in X(T) \\ \lambda+\nu \in W_p\cdot \mu} } \dim M_\nu \, \chi(w\cdot(\lambda + \nu))   }
,\end{align*}
where we sum over all weights linked to \(\mu\)

\begin{lemma}[Technical]

Let
\(\lambda,\mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
and \(\nu_1 \in X(T)_+ \cap W(\mu-\lambda)\). Then

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(\lambda + w\nu \not\in W_p\cdot \mu\) for any \(w\in W\) and
  \(\nu < \nu_1\).
\item
  If \(w\in W\) and \(\lambda + w\nu_1 \in W_p \cdot \mu\) (which can
  happen), then there exists some \(w_1\in W_p\) such that
\end{enumerate}

\begin{itemize}
\tightlist
\item
  \(w_1\cdot \lambda = \lambda\), so it stabilizes \(\lambda\),
\item
  \(w_1\cdot \mu = \lambda + w\nu_1\)
\end{itemize}

\end{lemma}

\hypertarget{characters-of-translated-modules}{%
\subsection{Characters of Translated
Modules}\label{characters-of-translated-modules}}

Goal: find \(\operatorname{ch}T_\lambda^\mu V\).

\begin{theorem}[?]

Let
\(\lambda, \mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
and \(V\) be a finite-dimensional module with
\({\operatorname{pr}}_\lambda V = V\). Write
\begin{align*}
\operatorname{ch}(V) = \sum_{w\in W_p} a_w \chi(w\cdot \lambda)
,\end{align*}
where \(a_w\in {\mathbb{Z}}\) and this is a finite sum. Then
\begin{align*}  
\operatorname{ch}(T_\lambda^\mu V) = 
\sum_{w\in W_p} a_w \qty{ 
  \sum_{w_1\in S} \chi(ww_1\cdot \mu)
},
\end{align*}
where \(S\) is a set of coset representatives for the group
\begin{align*}
{\operatorname{Stab}}_{W_p}(\lambda) \over {\operatorname{Stab}}_{W_p}(\mu) \cap {\operatorname{Stab}}_{W_p}(\lambda)
\end{align*}

\end{theorem}

\begin{proof}

We can write
\begin{align*}  
\operatorname{ch}(T_\lambda^\mu V)
&= \operatorname{ch}\qty{{\operatorname{pr}}_\mu\qty{L(\nu_1) \otimes{\operatorname{pr}}_\lambda V } } \\
&= \operatorname{ch}\qty{{\operatorname{pr}}_\mu\qty{L(\nu_1) \otimes V } } \\
&= \sum_{w\in W_p} a_w \qty{\sum_\nu \dim L(\nu_1)_\nu \chi(w\cdot(\lambda + \nu)) }\\
&= \sum_{w\in W_p} a_w  \qty{ \sum_\nu \dim L(\nu_1)_\nu \chi(w\cdot\lambda + \nu)  }
.\end{align*}

We need \(\lambda + \nu\in W_p\cdot \mu\) and \(\nu \leq \nu_1\) to
apply the technical lemma.

\begin{figure}
\centering
\includegraphics[width=2.60417in,height=\textheight]{figures/image_2020-10-26-14-13-36.png}
\caption{Inner and outershell, orbit of \(W\) action?}
\end{figure}

The last step can be written because the only contributions are
\(\nu \in W\nu_1\) and \(\dim L(\nu_1)_\nu = 1\), i.e.~we're on the
outer shell in the figure above.

We can apply (a) and (b) from the technical lemma to write
\begin{align*}  
\cdots 
&= \sum_{w\in W_p} a_w \sum_{w_1} \chi(w(w_1\cdot \mu))
.\end{align*}

By (b), \(w_1 \in {\operatorname{Stab}}_{W_p}(\lambda)\). We don't want
duplication, so we can check that
\(w_1\cdot\mu = w_2 \cdot\mu = \lambda+ w\nu_1\) implies that
\(w_1 \in w_2 {\operatorname{Stab}}_{W_p}(\mu)\). Thus we need to take
the coset representatives stated in the theorem.

\begin{quote}
Thus we don't need to consider any weights in the inner shell.
\end{quote}

\end{proof}

\hypertarget{equivalence-of-categories}{%
\subsection{Equivalence of Categories}\label{equivalence-of-categories}}

Goal: show that a pair of functors each admit a natural transformation
to the identity.

\begin{definition}[Natural transformations, isomorphisms, and equivalence of categories]

Let \(\mathcal{C}, \mathcal{D}\) be categories and
\(S, T:\mathcal{C} \to \mathcal{D}\) be two functors. A \textbf{natural
transformation} \(\alpha:S\to T\) is a function that assigns to each
object \(c\in \mathcal{C}\) a morphism \(\alpha_c:S(c) \to T(c)\) in
such a way that for every \(f:c\to c'\), we have a commuting square

\begin{center}
\begin{tikzcd}
S(c) \ar[r, "{\alpha_c}"] \ar[d, "S(f)"] & T(c)\ar[d, "T(f)"] \\
S(c') \ar[r, "{\alpha_c}"] & T(c')
\end{tikzcd}
\end{center}

If \(\alpha_c\) is an equivalence for all \(c\in \mathcal{C}\), then
\(\alpha\) is said to be a \textbf{natural isomorphism}. Two categories
are said to be \textbf{equivalent} iff \(S\circ T\) and \(T\circ S\) are
naturally isomorphic to the identity functor.

\end{definition}

\begin{theorem}[?]

Suppose
\(\lambda, \mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
belong to the same facet. Then \(T_\lambda^\mu\) induces an equivalence
of categories from
\begin{align*}  
\mathcal{B}_\lambda \coloneqq\mathcal{C} \coloneqq\left\{{V\in {\mathsf{G}{\hbox{-}}\mathsf{Mod}} {~\mathrel{\Big|}~}{\operatorname{pr}}_\lambda V = V}\right\} \leadsto
\mathcal{B}_\mu \coloneqq\mathcal{D} \coloneqq\left\{{V\in {\mathsf{G}{\hbox{-}}\mathsf{Mod}} {~\mathrel{\Big|}~}{\operatorname{pr}}_\mu V = V}\right\}
.\end{align*}
where
\(T_\mu^\lambda \circ T_\lambda^\mu \cong {\operatorname{pr}}_\lambda\).

\end{theorem}

\begin{proof}

Using the adjointness of \(T_\lambda^\mu\) and
\(T_\mu^\lambda\)\textless{} we can write
\begin{align*}  
\hom_G(V, T_\mu^\lambda T_\lambda^\mu V) \equiv
\hom_G(T_\lambda^\mu V,  T_\lambda^\mu V)
.\end{align*}

So consider the identity map on the latter
\(\operatorname{id}: T_\lambda^\mu V{\circlearrowleft}\), and let
\(f_V: V\to T_\mu^\lambda T_\lambda^\mu V\) be the corresponding map in
the former. We can a natural transformation in the following way

\begin{center}
\begin{tikzcd}
V \ar[r, "{f_V}"]\ar[d, "\operatorname{id}"] & T_\mu^\lambda T_\lambda^\mu V \ar[d, "\operatorname{id}'"] \\
V' \ar[r, "{f_{V'}}"] & T_\mu^\lambda T_\lambda^\mu V'
\end{tikzcd}
\end{center}

It suffices to show that the \(f_V\) are isomorphism as maps of
\(G{\hbox{-}}\)modules, so one proceeds by

\begin{itemize}
\item
  Showing it works for simple \(G{\hbox{-}}\)modules, and
\item
  Applying induction to composition length, using the five lemma.
\end{itemize}

Suppose \(V\) is simple, then by the prior theorem we can write
\begin{align*}  
\operatorname{ch}T_\mu^\lambda T_\lambda^\mu V = 
\sum_{w\in W_p} a_w \qty{\sum_{w_1, w_2} \chi\qty{w(w_2 w_1)\cdot \lambda } }
.\end{align*}
We know that

\begin{itemize}
\tightlist
\item
  \(w_1\in {\operatorname{Stab}}_{W_p}(\mu) / \sim\)
\item
  \(w_2\in {\operatorname{Stab}}_{W_p}(\lambda) / \sim\)
\end{itemize}

Note that if \(\mu, \lambda\) are in the same facet, then the
stabilizers are the same.

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-26-14-42-57.png}
\caption{Weights in the same facet share a stabilizer.}
\end{figure}

So \(\lambda, \mu\) are in the same facet, so
\(w_1 = \operatorname{id}, w_2 = \operatorname{id}\) and \(f_V\) is an
isomorphism. We thus obtain
\(T_\mu^\lambda T_\lambda^\mu =\cong {\operatorname{pr}}_\lambda\) and
\(T_\lambda^\mu T_\mu^\lambda =\cong {\operatorname{pr}}_\mu\). Thus
\(B_\lambda \cong B_\mu\).

\end{proof}

\emph{Question}: What happens when translating from an alcove onto a
wall? A similar formula will hold in this case: we will get either
induced modules or zero, depending on the dominance of the weights. This
will lead into the Lusztig conjectures.

\hypertarget{wednesday-october-28}{%
\section{Wednesday, October 28}\label{wednesday-october-28}}

\hypertarget{review-of-last-time}{%
\subsection{Review of Last Time}\label{review-of-last-time}}

Suppose we have two weights in the same facet, i.e.~they're in the same
stabilizer under the action of the affine Weyl group:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-28-13-56-17.png}
\caption{Weights in the same facet}
\end{figure}

We had a theorem: if \(\lambda, \mu\) are in the same facet, then
\(\mathcal{B}_\lambda \cong \mathcal{B}_\mu\) is an equivalence of
categories, where the map is via the translation functors.

\hypertarget{description-of-t_lambdamu-hiwcdot-lambda}{%
\subsection{\texorpdfstring{Description of
\(T_\lambda^\mu {H^i(w\cdot \lambda) }\)}{Description of T\_\textbackslash lambda\^{}\textbackslash mu \{H\^{}i(w\textbackslash cdot \textbackslash lambda) \}}}\label{description-of-t_lambdamu-hiwcdot-lambda}}

We can write
\begin{align*}  
T_\lambda^\mu \qty{H^i(w\cdot \lambda)} 
&= {\operatorname{pr}}_\mu \qty{L(\nu_1) \otimes{\operatorname{pr}}_\lambda\qty{H^i(w\cdot \lambda)} } \\
&= {\operatorname{pr}}_\mu \qty{L(\nu_1) \otimes{H^i(w\cdot \lambda)} } \\
&= {\operatorname{pr}}_\mu \qty{L(\nu_1) \otimes{R^i \mathop{\mathrm{ind}}_B^G w\cdot \lambda} } \\
&= {\operatorname{pr}}_\mu\qty{R^i \mathop{\mathrm{ind}}_B^G \qty{L(\nu_1) \otimes w\cdot \lambda } }
.\end{align*}

Take a composition series by \(B{\hbox{-}}\)modules of
\(L(\nu_1) \otimes w\cdot \lambda\), say
\begin{align*}  
0 = M_0 \subseteq M_1 \cdots \subseteq M_r = L(\nu_1) \otimes w\cdot \lambda
.\end{align*}
where \(M_j / M_{j-1} \cong \lambda+j + w\cdot \lambda\) and
\(\lambda_j < \lambda_{j'} \implies j < j'\), i.e.~we can order them in
a decreasing way.

Consider the SES

\begin{center}
\begin{tikzcd}
0 \ar[r] & M_{j-1} \ar[r] & M_j \ar[r] & M_{j} / M_{j-1} \ar[r] & 0
\end{tikzcd}
\end{center}

where applying \({\operatorname{pr}}_\mu({-})\) induces the LES

\begin{center}
\begin{tikzcd}
\cdots \ar[r] & {\operatorname{pr}}_\mu M_{j-1} \ar[r] & {\operatorname{pr}}_\mu M_j \ar[r] & {\operatorname{pr}}_\mu \left( M_{j} / M_{j-1} \right) \ar[r] & \cdots
\end{tikzcd}
\end{center}

We know that
\begin{align*}  
{\operatorname{pr}}_\mu H^i\qty{\lambda_j + w\cdot \lambda} = 
\begin{cases}
H^i(\lambda_j + w\cdot \lambda ) & \lambda+j + w\cdot \lambda \in W_p\cdot \mu \\
0 & \text{else}
\end{cases}
,\end{align*}
i.e.~this projection is the identity for weights linked to \(\mu\) and
zero otherwise. We also have
\begin{align*}  
{\operatorname{pr}}_\mu H^i(M_r) = T_\lambda^\mu H^i(w\cdot \lambda)
.\end{align*}

\begin{theorem}[?]

Let
\(\lambda, \mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
and \(F\) be a facet with \(\lambda \in F\). If
\(\mu \in \mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu\),
then we have
\begin{align*}  
T_\lambda^\mu\qty{H^i(w\cdot \lambda)} = H^i(w\cdot \mu) \qquad \forall w\in W_p
.\end{align*}

\end{theorem}

\begin{example}[?]

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-28-14-12-31.png}
\caption{Image}
\end{figure}

Here consider \(H_0(\lambda) \xrightarrow{T_\lambda^\mu} H_0(\mu) = 0\),
since \(\mu\) is outside of the dominant region (in orange.) We also
have \(H^0(w\cdot \lambda) \to H^0(w\cdot \mu) \neq 0\), since this
falls \emph{into} the dominant region.

\end{example}

\begin{proof}[?]

Let \(\lambda \in F\) and
\(\mu\in\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu\).
Then
\({\operatorname{Stab}}_{W_p}(\lambda) \subseteq {\operatorname{Stab}}_{W_p}(\mu)\).
By a previous technical lemma, we had a formula for computing
\(\operatorname{ch}T_\lambda^\mu V\), which involved considering
\begin{align*}  
w_1 \in {{\operatorname{Stab}}_{W_p}(\lambda) \over {\operatorname{Stab}}_{W_p}(\lambda) \cap{\operatorname{Stab}}_{W_p}(\mu)}
.\end{align*}
In this case, we get \(w_1 = \operatorname{id}\), since the top and
bottom are equal.

By that lemma, there exists a unique \(\ell\) such that
\(w\cdot \lambda + \lambda_\ell \in W_p\cdot \mu\), where
\(\lambda_\ell\) is a weight of \(L(\nu_1)\). From the LES, we have

\begin{center}
\begin{tikzcd}
\cdots \ar[r] & {\operatorname{pr}}_\mu M_{j-1} \ar[r] & {\operatorname{pr}}_\mu M_j \ar[r] & {\operatorname{pr}}_\mu \left( M_{j} / M_{j-1} \right) = \lambda_j + w\cdot \lambda \ar[r] & \cdots
\end{tikzcd}
\end{center}

where the last term will only be nonzero in restricted cases. We can
thus conclude that
\begin{align*}  
{\operatorname{pr}}_\mu(H^i(M_j))  =
\begin{cases}
0 & j< \ell \\
H^i(w\cdot \mu) & j\geq \ell.
\end{cases}
\end{align*}
Setting \(j=r\), we have
\begin{align*}  
T_\lambda^\mu \qty{H^i(w\cdot\lambda)} = {\operatorname{pr}}_\mu H^j(M_r) = H^i(w\cdot \mu)
.\end{align*}

\end{proof}

Suppose
\(\lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
and \(\mu \in C_{\mathbb{Z}}\). What happens when you translate
\(\lambda\) (blue) off of a wall?
\(T_\lambda^\mu\qty{H^0(w\cdot \lambda)}\) has a filtration with factors
\(H^0(w_1\cdot \mu)\) and \(H^0(w_2\cdot \mu)\) (shown in green).

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-10-28-14-25-12.png}
\caption{Filtration coming from translating off of a wall}
\end{figure}

If \(w\lambda\) is a vertex with \(\mu \in C_{\mathbb{Z}}\), then
\(T_\lambda^\mu(H^0(w\cdot \lambda))\) has six factors:

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-10-28-14-27-05.png}
\caption{Weight where the translation has six factors}
\end{figure}

\begin{proposition}[?]

Suppose
\(\lambda \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
and \(\mu \in C_{\mathbb{Z}}\), and let \(w\in W_p\) where
\(w\cdot \lambda \in X(T)_+\). Then
\(T_\lambda^\mu (H^0(w\cdot \lambda))\) has a filtration such that all
of the composition factors are of the form \(H^0(ww_1 \cdot \mu)\) where
\(w_1\in {\operatorname{Stab}}_{W_p}(\lambda)\) and each of the factors
occurs at most once.

\end{proposition}

Recall that \(\widehat{F}\) denotes the \emph{upper closure}.

\begin{proposition}[?]

Let
\(\lambda, \mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)
be in the bottom alcove, where
\(\mu \in \mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu_1\)
but \(\lambda\in F_1\). Let \(F\) be the facet containing
\(w\cdot\lambda\), then
\begin{align*}  
T_\lambda^\mu(L(w\cdot \lambda)) = 
\begin{cases}
L(w\cdot \mu)  & w\cdot \mu \in \widehat{F} \\
0 & \text{else}.
\end{cases}
\end{align*}

\end{proposition}

\begin{example}[?]

In this situation, we have \(T_\lambda^\mu(L(\lambda)) = 0\):

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-10-28-14-35-11.png}
\caption{Image}
\end{figure}

If instead \(\mu \in \widehat{C}_{\mathbb{Z}}\), we have
\(T_\lambda^\mu( L(\lambda)) = L(\mu)\):

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-10-28-14-36-02.png}
\caption{Image}
\end{figure}

\end{example}

\textbf{Big Question}: What happens to \(L(w\cdot \lambda)\) when
translating away from a wall?

\begin{definition}[Walls]

A facet \(F\) is a \textbf{wall}
\(\iff {\left\lvert { \Phi_0^+(F) } \right\rvert} = 1\). In this case,
there exists a unique \(\alpha\) such that
\({\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = n_\alpha p\).

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-28-14-39-23.png}
\caption{Example of a wall}
\end{figure}

\end{definition}

\begin{remark}

Note that \(s_F = s_{\beta, n_p}\) where
\(n_p = {\left\langle {\lambda + \rho},~{\beta {}^{ \vee }} \right\rangle}\)
acts on the wall as the identity and reflects across it:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-28-14-41-36.png}
\caption{Image}
\end{figure}

Here \({\operatorname{Stab}}_{W_p}(\lambda) = \left\{{1, s_F}\right\}\).

\end{remark}

\begin{proposition}[?]

Consider the following situation:

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-28-14-43-39.png}
\caption{Image}
\end{figure}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \([T_\mu^\lambda (L(w\cdot \mu)) : L(w\cdot \lambda)] = 2\), appearing
  once in the socle and once in the head.
\item
  \(L(w\cdot \lambda) = \mathop{\mathrm{Soc}}_G T_\mu^\lambda(L(w\cdot \mu)) = T_\mu^\lambda(L(w\cdot \mu)) / \mathop{\mathrm{Rad}}T_\mu^\lambda (L(w\cdot \mu))\).
\end{enumerate}

\begin{figure}
\centering
\includegraphics{figures/image_2020-10-28-14-46-57.png}
\caption{Heart of the module}
\end{figure}

\end{proposition}

\textbf{Big Problems}:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  When is the heart semisimple?
\item
  Determine the composition factors in the heart?
\end{enumerate}

Given these, you could compute dimensions of irreducible
representations.

\hypertarget{monday-november-02}{%
\section{Monday, November 02}\label{monday-november-02}}

Today: Lusztig conjectures, but first some alcove geometry.

\hypertarget{alcove-geometry}{%
\subsection{Alcove Geometry}\label{alcove-geometry}}

\hypertarget{length-function-for-alcoves}{%
\subsubsection{Length Function for
Alcoves}\label{length-function-for-alcoves}}

Let \(A, B\) be alcoves, and recall that a hyperplane is give by
\begin{align*}  
H_{\alpha, m} \coloneqq\left\{{ \lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} = mp }\right\}
.\end{align*}

These are codimension 1 objects:

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-02-13-55-24.png}
\caption{Image}
\end{figure}

We can also divide these into positive and negative sides:

\begin{align*}  
H_{\alpha, m}^+ &\coloneqq\left\{{   \lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} > mp }\right\} \\
H_{\alpha, m}^- &\coloneqq\left\{{   \lambda \in X(T) {~\mathrel{\Big|}~}{\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} < mp }\right\} 
.\end{align*}

Let \(S(A, B)\) be the set of hyperplanes separating \(A\) and \(B\).

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-02-13-58-08.png}
\caption{Hyperplanes separating two alcoves}
\end{figure}

If \(H\in S(A, B)\), then define a function
\begin{align*}  
{\varepsilon}(H) \coloneqq
\begin{cases}
1 & A\in H_{\alpha, m}^- \\
-1 & A\in H_{\alpha, m}^+
\end{cases}
,\end{align*}
and from it construct a \textbf{distance function}
\begin{align*}  
d(A, B) \coloneqq\sum_{H\in S(A, B)} {\varepsilon}(H)
.\end{align*}

Recall that we can define
\begin{align*}  
C_{\mathbb{Z}}= \left\{{\lambda \in X(T) {~\mathrel{\Big|}~}0 < {\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle}<p \,\, \forall \alpha\in\Phi^+ }\right\}
,\end{align*}
and for \(\lambda \in C_{\mathbb{Z}}\)

\begin{align*}  
d(w\cdot \lambda) = d(C_{\mathbb{Z}}, wC_{\mathbb{Z}})
.\end{align*}

for \(w\in W_p\).

Now consider the following situation:

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-02-14-02-44.png}
\caption{Image}
\end{figure}

Recall that computing \(\operatorname{ch}L(\lambda)\) is equivalent to
finding \([H^0(w\cdot \lambda): L(w_2\cdot \lambda)]\). We know that
\begin{align*}  
[T_\mu^\lambda (L(w\cdot\mu)) : L(w\cdot \mu)] = 2
,\end{align*}
where the \(w\)s are now the same, and the structure of the module is

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-02-14-06-10.png}
\caption{O}
\end{figure}

where knowing the structure of the heart of the module is an open
problem.

The setup is the following: Let \(\lambda\in X(T) \cap C_{\mathbb{Z}}\)
where
\(\mu \in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\).
Since it's on a wall, there is a nontrivial stabilizer
\({\operatorname{Stab}}_{W_p}(\mu) = \left\langle{s}\right\rangle\). We
can write
\begin{align*}  
\operatorname{ch}L(w\cdot \lambda) = \sum_{w'} a_{w, w'} \chi(w' \cdot \lambda)
.\end{align*}

The sum is over \(w'\in W_p\) where \(w'\cdot \lambda \in X(T)_+\), and
we have strong linkage \(w'\cdot \lambda \uparrow w\cdot \lambda\). Now
consider translating from \(\lambda\) to the wall \(\mu\). We can write
\begin{align*}  
\operatorname{ch}L(w\cdot \lambda) = \sum_{w'} a_{w, w'} \chi(w' \cdot \mu)
.\end{align*}

Since \(\left\langle{s}\right\rangle\) is the stabilizer, we have
\begin{align*}  
\operatorname{ch}T_\mu^\lambda L(w\cdot \mu)
&= \sum_{w'} a_{w, w'} \chi(w' \cdot \lambda)
+ \sum_{w'} a_{w, w'} \chi(w's \cdot \mu) \\
&= \operatorname{ch}L(w\cdot \lambda)
+ \sum_{w'} a_{w, w'} \chi(w's \cdot \mu)
,\end{align*}

We can now check that
\begin{align*}  
\operatorname{ch}T_\mu^\lambda T_\lambda^\mu L(w\cdot \lambda) = 
\operatorname{ch}L(w\cdot \lambda) + \chi(ws\cdot \lambda ) + \sum_{w'} b_{w'} \chi(w'\cdot\lambda)
.\end{align*}

Note that in the last sum, the \(b_{w'}\) with \(w'\in W_p\) satisfy
\(w'\cdot \lambda \in X(T)_+\) and
\(d(w'\cdot\lambda) < d(w\cdot \lambda)\). We would like to compute
\begin{align*}  
[H^0(w_1\cdot \lambda) : L(w_2\cdot \lambda)
.\end{align*}
using induction on \(d(w_1\cdot \lambda)\).

Suppose that for all \(w_1\) with
\(d(w_1\cdot\lambda) < d(w\cdot \lambda)\), we know all of the
\(b_{w'}\). In this case, we know \(\chi(ws\cdot \lambda)\) by Weyl's
character formula. We'd then only need to know the translated character
\begin{align*}  
T_\mu^\lambda T_\lambda^\mu L(w\cdot \lambda)
.\end{align*}
Note that it's sufficient to know all of its composition factors
\begin{align*}  
[T_\mu^\lambda T_\lambda^\mu L(w\cdot \lambda): L(w_2\cdot \lambda)
,\end{align*}

and since we know the head and the socle, it suffices to understand the
composition factors in its heart:

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-02-14-20-41.png}
\caption{Understanding the composition factors of the heart.}
\end{figure}

\begin{remark}[Question (Vogan, Beilinson-Bernstein)]

Is the heart semisimple for
\begin{align*}  
{\left\langle {w(\lambda + \rho)},~{\alpha {}^{ \vee }} \right\rangle} \leq p(p-h-2) && \forall \alpha\in \Phi^+
.\end{align*}
where \(w\cdot\lambda = w(\lambda + \rho) - \rho\)?

\end{remark}

\begin{definition}[Coxeter Number]

Define \(h\) to be the \textbf{Coxeter number}, defined by
\begin{align*}  
h \coloneqq{\left\langle {\rho},~{\alpha_0 {}^{ \vee }} \right\rangle} + 1
\end{align*}
where \(\alpha_0\) is the highest short root.

\end{definition}

\begin{example}[?]

Let \(\Phi = A_n\), then
\(\alpha_0 = \alpha_1 + \alpha_2 + \cdots + \alpha_n\). Then \(h=n+1\).

\end{example}

\begin{example}[?]

Let \(\Phi = G_2\), then \(\alpha_n = 3\alpha_1 + 2\alpha_2\), and
\(h = 5+1 = 6\).

\end{example}

\begin{remark}

Andersen showed that this question is equivalent to the Lusztig
conjecture. To state this conjecture, we'll need a few more definitions.

\end{remark}

\begin{definition}[Jantzen Region]

Define the \textbf{Jantzen region} by
\begin{align*}  
\operatorname{Jan} \coloneqq\left\{{\lambda \in X(T) {~\mathrel{\Big|}~}
0 \leq {\left\langle {\lambda + \rho},~{\alpha {}^{ \vee }} \right\rangle} \leq p(p-h-2) \,\, \forall \alpha\in \Phi^+
}\right\}
.\end{align*}

\end{definition}

\begin{proposition}[Lusztig Conjecture]

Let \(\lambda \in X(T) \cap C\) and
\(w\cdot\lambda \in \operatorname{Jan}\) with \(w\in W_p\). Then
\begin{align*}  
\operatorname{ch}L(w\cdot \lambda) = \sum_{w'} (-1)^{d(w\cdot \lambda) - d(w'\cdot \lambda)} P_{w'w_0, ww_0} (1) \chi(w'\cdot \lambda)
.\end{align*}
where \(w'\in W_p\) with \(w'\cdot\lambda \in X(T)_+\). The polynomials
\(P\) appearing here are the KL-polynomials for \(W_p\) and \(w_0\) is
the longest element in \(W\).

\end{proposition}

\begin{remark}

Note that the Lusztig conjecture is similar to the Kazhdan-Lusztig
conjectures: in that setting, we're in category \({\mathcal{O}}\), we
have the Weyl group instead of the affine Weyl group, and we're
concerned with the characters of Verma modules. Note that this
conjecture is true for category \({\mathcal{O}}\) and in the case of
quantum groups. In the quantum group setting, you can only iterate
Frobenius once, but for algebraic groups it can be iterated. The
presence of the affine Weyl group also makes these settings very
different.

\end{remark}

\begin{remark}

Some remarks on the conjecture:

\begin{itemize}
\item
  This is true for \(p\gg 0\), by Andersen-Jantzen-Soergel (1994) and
  Kazhdan-Lustzig, Koshiwana-Tanisaki.
\item
  It was unknown if there was an effective lower bound until it was
  found by Fiebig (possibly in 2000s)
\item
  Williamson demonstrated a counterexample by looking at Steinberg
  weights.
\end{itemize}

\end{remark}

If we know \(\operatorname{ch}L(\lambda)\) for \(\lambda \in X_1(T)\) in
the restricted region, by Steinberg's twisted tensor product formula we
can compute \(L(\mu)\) for \(\mu \in X(T)_+\). So when is the restricted
region contained in the Jantzen region, so
\(X_1(T) \subseteq \operatorname{Jan}\)? Assume that
\(\lambda \in X_1(T)\), then compute
\begin{align*}  
{\left\langle {\lambda+\rho},~{\alpha_0 {}^{ \vee }} \right\rangle} = {\left\langle {\lambda},~{\alpha_0 {}^{ \vee }} \right\rangle} + (h-1)
.\end{align*}
Note that by extending scalars to \({\mathbb{Q}}\), we get an inequality
in the ordering of the form \(\lambda \leq_{\mathbb{Q}}(p-1_\rho)\). We
can then write

\begin{align*}  
{\left\langle {\lambda+\rho},~{\alpha_0 {}^{ \vee }} \right\rangle} 
&= {\left\langle {\lambda},~{\alpha_0 {}^{ \vee }} \right\rangle} + (h-1) \\
\leq {\left\langle {(p-1) \rho},~{\alpha_0 {}^{ \vee }} \right\rangle} + (h-1)  \\
&= (p-1)(h-1) + (h-1) \\
&= p(h-1) 
.\end{align*}
When is this less than or equal to \(p(p-h-2)\)? We can check that this
happens iff
\begin{align*}  
p(h-1) &\leq p(p-h-2) \\
\iff h-1 &\leq p-h-2 \\
\iff 2h+1 &\leq p
,\end{align*}
so the conjecture (as stated in its original form) reads
\begin{align*}  
p\geq 2h+1 \implies X_1(T) \subseteq \operatorname{Jan}
.\end{align*}

\begin{remark}

Kato conjectured that this bound could be lowered to \(p\geq h\), but
Williamson produced counterexamples around 2013. Essentially, the
calculation of \(\operatorname{ch}L(\lambda)\) is still open.

\end{remark}

Next time: we'll define \(G_rT{\hbox{-}}\)modules. We'll define a graded
category to prevent the weights from collapsing mod \(p\). These turn
out to be easier to work with than \(G_r{\hbox{-}}\)modules, and results
can be pushed down.

\hypertarget{wednesday-november-04}{%
\section{Wednesday, November 04}\label{wednesday-november-04}}

Today: \(G_r{\hbox{-}}T\) modules.

Note that \(G_r{~\trianglelefteq~}G_r T\), with
\(G_r T/G_r \cong T^{(r)}\). We consider \(G_r T{\hbox{-}}\)modules,
which are \(G_r{\hbox{-}}\)modules with a \(T\) action given by
\begin{align*}  
G_r \times M &\to M \\
(g, m) &\mapsto g\cdot m
\end{align*}
which are \(T{\hbox{-}}\)equivariant,
i.e.~\(t(g\cdot m) = (t\cdot g)(t\cdot m)\) for \(t\in T, g\in G_r\),
and \(m\in M\) is a \(G_r T{\hbox{-}}\)module. This essentially induces
a grading on \(G_r\).

\hypertarget{representations-for-g_r-t-and-g_r-b}{%
\subsection{\texorpdfstring{Representations for \(G_r T\) and
\(G_r B\)}{Representations for G\_r T and G\_r B}}\label{representations-for-g_r-t-and-g_r-b}}

Recall that we have a Frobenius map, for which we take the
scheme-theoretic kernel:
\begin{align*}  
F&: G \to G \\ 
F^r &\coloneqq F\circ F \circ \cdots \circ F \\
G_r &\coloneqq\ker F^r
,\end{align*}
and we then define
\begin{align*}  
G_r T \coloneqq(F^r)^{-1} (T) \\
G_r B \coloneqq(F^r)^{-1} (B) 
\end{align*}
taken as scheme-theoretic objects.

Noting that \(B\subset G_r B\), for \(\lambda \in X(T)\) we define
\begin{align*}  
\widehat{Z}_r' (\lambda) &\coloneqq\mathop{\mathrm{ind}}_B^{G_r B} \lambda \\
\widehat{Z}_r (\lambda) &\coloneqq\mathop{\mathrm{coInd}}_B^{G_r B} \lambda
.\end{align*}

These are enhancements of the baby Verma modules, in the sense that if
we take restrictions we get
\begin{align*}  
\widehat{Z}_r' (\lambda) \downarrow_{G_r} = \mathop{\mathrm{ind}}_{B_r}^{G_r} \lambda
.\end{align*}

We similarly have
\begin{align*}  
{Z}_r' (\lambda) \downarrow_{G_r T} &= \mathop{\mathrm{ind}}_{B_r T}^{G_r T} \lambda \\
\widehat{Z}_r' (\lambda) \downarrow_{G_r T} &= \mathop{\mathrm{coInd}}_{B_r T}^{G_r T} \lambda
.\end{align*}

\begin{proposition}[?]

\envlist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\widehat{Z}_r(\lambda + p^r \mu) \cong \widehat{Z}_r(\lambda) \otimes p^r \mu\)
\item
  \(\widehat{Z}_r' (\lambda + p^r \mu) \cong \widehat{Z}_r' (\lambda) \otimes p^r \mu\)
\item
  \(\operatorname{ch}\widehat{Z}_r(\lambda) = \operatorname{ch}\widehat{Z}_r' (\lambda) = e^\lambda \prod_{\alpha\in\Phi^+} {1 - e^{-p^r\mu} \over 1 - e^{-\alpha}}\).
\end{enumerate}

\end{proposition}

\begin{proof}[of 1 and 2]

From the definition, we have
\begin{align*}  
\widehat{Z}_r'(\lambda + p^r \mu)
&= \mathop{\mathrm{ind}}_{B}^{G_r B} (\lambda + p^r \mu) \\
&= \mathop{\mathrm{ind}}_{B}^{G_r B} (\lambda \otimes p^r \mu) \\
\cong \qty{\mathop{\mathrm{ind}}_B^{G_r B} \lambda} \otimes p^r\mu
.\end{align*}
Where in the last equality we've applied the tensor identity, noting
that \(p^r\mu\) is a 1-dimensional \(G_r B{\hbox{-}}\)module, since
\begin{align*}  
G_r B \to G_r B/G_r = B^{(r)} = B/B_r
,\end{align*}
making it a representation by pullback.

\end{proof}

\begin{proof}[of 3]

We can write
\begin{align*}  
\widehat{Z}_r(\lambda) = \operatorname{dist}(U_r) \otimes\lambda
,\end{align*}
and thus
\begin{align*}  
\operatorname{ch}\widehat{Z}_r(\lambda) 
&= e^{\lambda} \operatorname{ch}\operatorname{dist}(U_r)  \\
&= e^{\lambda} \prod_{\alpha\in \Phi^+}\qty{ 1 + e^{-\alpha} + \cdots + e^{-(p^r-1)\alpha} } \\
&= e^\lambda {1 - e^{-p^r \alpha} \over 1 - e^{-\alpha}} && \text{as a geometric series}
.\end{align*}

\end{proof}

The next theorem is related to the fact that when comparing these
categories of modules, one is essentially a graded version of the other.

\begin{theorem}[?]

Let \(M\in{\mathsf{G_rT}{\hbox{-}}\mathsf{Mod}}\), then TFAE:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(M\) is an injective \(G_r T{\hbox{-}}\)module.
\item
  \(M\) is an injective \(G_r{\hbox{-}}\)module.
\end{enumerate}

\end{theorem}

Note that \(G_r {~\trianglelefteq~}G_r T\), where the quotient is
\(T^{(r)}\) which is twisted by Frobenius \(r\) times.

\begin{proof}[?]

We'll apply the Lydon-Hoschild-Serre spectral sequence: for \(N\) a
\(G_r T{\hbox{-}}\)module,
\begin{align*}  
E_2^{i, j} = \operatorname{Ext} _{T^{(r)}}^{i}\qty{K, \operatorname{Ext} _{G_r}^j\qty{N, M} } \Rightarrow\operatorname{Ext} _{G_r T}^{i+j}(N, M)
.\end{align*}

\(2\implies 1\):

We first note that 2 implies
\(\operatorname{Ext} _{G_r}^{>0}(N, M) = 0\), so the spectral sequence
collapses and we have
\begin{align*}  
\operatorname{Ext} _{T^{(r)}}^i (k, \hom_{G_r}(M, N) ) \cong \operatorname{Ext} _{G_r T}^i(N, M)
.\end{align*}
Since modules over \(T^{(r)}\) are completely reducible, we have
\begin{align*}  
\operatorname{Ext} _{T^{(r)}}^{>0} ( k, \hom_{G_r}(N, M) ) = 0
,\end{align*}

and thus \(\operatorname{Ext} _{G_r T}^{>0}(N, M) = 0\), making \(M\) an
injective \(G_r T{\hbox{-}}\)module. \(\hfill\blacksquare\)\\

\(1\implies 2\):

The simple \(G_r T{\hbox{-}}\)modules are of the form
\(N\coloneqq L_r(\lambda) \otimes p^r\sigma\) where
\(\lambda\in X_r(T)\) and \(\sigma\in X(T)\). Note that \(L_r(\lambda)\)
is simple \(G_r{\hbox{-}}\)module. Applying the spectral sequence, there
is a 5 term exact sequence. Letting
\(E_t \coloneqq\operatorname{Ext} _{G_rT}^t (N, M)\).

\begin{center}
\begin{tikzcd}
0 \ar[r] &
E_2^{1, 0} \ar[r] &
E_1         \ar[r] &
E_2^{0, 1} \ar[r] &
E_2^{2, 0} \ar[r] &
E_2
\end{tikzcd}
\end{center}

Everything is zero here except for the middle term: \(E_1, E_2 = 0\) by
assumption? \(E_2^{1, 0}, E_2^{2, 0} = 0\) by ?.

We can thus conclude that
\begin{align*}  
0 = E_2^{0, 1}
&= \hom_{T^{(r)}}(k, \operatorname{Ext} _{G_r}^1 (L_r(\lambda) \otimes p^r\sigma, M)  ) \\ 
&= \hom_{T^{(r)}}(p^r\sigma, \operatorname{Ext} _{G_r}^1 (L_r(\lambda), M)  ) 
,\end{align*}
which holds for all \(p^r \sigma\), and thus
\(\operatorname{Ext} _{G_r}^1 (L_r(\lambda), M) = 0\) for all
\(\lambda\in X_1(T)\). So \(M\) is injective as a
\(G_r{\hbox{-}}\)module.

\end{proof}

\begin{proposition}[?]

Let \(\lambda \in X(T)\), then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(\widehat{Z}_r (\lambda) \downarrow_{B_r T}\) is the projective cover
  of \(\lambda\) and the injective hull of \(\lambda - 2(p^r-1)\rho\).
\item
  \(\widehat{Z}_r' (\lambda) \downarrow_{B_r^+}\) is the projective
  cover of \(\lambda - 2(p^r-1)\rho\) and the injective hull of
  \(\lambda\).
\end{enumerate}

\end{proposition}

\hypertarget{summary-classification-of-simple-g_r-thbox-modules}{%
\subsection{\texorpdfstring{Summary: Classification of Simple
\(G_r T{\hbox{-}}\)Modules}{Summary: Classification of Simple G\_r T\{\textbackslash hbox\{-\}\}Modules}}\label{summary-classification-of-simple-g_r-thbox-modules}}

\begin{itemize}
\tightlist
\item
  \(\mathop{\mathrm{Soc}}_{B_r^+} \widehat{Z}_r '(\lambda) = \lambda\)
\item
  \(\widehat{Z}_r ' (\lambda) ^{U^+} = \lambda\), where the RHS denotes
  the \(U^+\) invariants.
\item
  Let
  \(\widehat{L}_r(\lambda) \coloneqq\mathop{\mathrm{Soc}}_{G_r T} \widehat{Z}_r' (\lambda)\).
\item
  Each simple \(G_r T{\hbox{-}}\)module is isomorphic to
  \(\widehat{L}_r (\lambda)\) for some \(\lambda\in X(T)\).
\item
  \(\widehat{L}_r(\lambda) \downarrow_{G_r} \cong L_r (\lambda)\) for
  all \(\lambda \in X_1(T)\).
\item
  Translation invariance:
  \(\widehat{L}_r(\lambda + p^r\sigma) = \widehat{L}_r (\lambda) \otimes p^r\sigma\)
\item
  \(\widehat{L}_r (\lambda + p^r \sigma) \downarrow_{G_r} = L_r(\lambda)\)
  for all \(\lambda \in X_r(T)\).
\end{itemize}

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-04-14-44-01.png}
\caption{Example in \(\Phi = B_2\)}
\end{figure}

\begin{quote}
This essentially allows you to replace working mod \(p\) in
characteristic \(p\) with working with integers instead, allowing the
usual weight theory to be used.
\end{quote}

\begin{proposition}[?]

Let \(\lambda \in X(T)\), then there exists an isomorphism of
\(G{\hbox{-}}\)modules
\begin{align*}  
H^i(\lambda) = R^i \mathop{\mathrm{ind}}_{G_r B}^G \widehat{Z}_r ' (\lambda)
,\end{align*}
where
\(\widehat{Z}_r'(\lambda) = \mathop{\mathrm{ind}}_B^{G_r B}(\lambda)\).

\end{proposition}

\hypertarget{friday-november-06}{%
\section{Friday, November 06}\label{friday-november-06}}

\hypertarget{good-p-rhbox--filtrations}{%
\subsection{\texorpdfstring{Good \((p, r){\hbox{-}}\)
Filtrations}{Good (p, r)\{\textbackslash hbox\{-\}\} Filtrations}}\label{good-p-rhbox--filtrations}}

Last time: \(G_r T\) and \(G_r B\) modules. We roughly know the category
of \(G_r\) modules, and we think of \(G_r T\) as graded
\(G_r{\hbox{-}}\)modules. We defined
\begin{align*}  
\widehat{Z}_r' (\lambda) &\coloneqq\mathop{\mathrm{ind}}_{B}^{G_r B}(\lambda) \\
\widehat{Z}_r' (\lambda) &\coloneqq\mathop{\mathrm{coInd}}_{B^+}^{G_r B^+}(\lambda)
.\end{align*}

We can use these for classification since we have a correspondence
\begin{align*}  
\left\{{\substack{\text{Simple }G_rT{\hbox{-}}\text{modules}}}\right\}
&\iff
\left\{{\substack{X(T)}}\right\} \\
\widehat{L}_r(\lambda) = \widehat{L}_r(\lambda_0) \otimes p^r \lambda &\mapsfrom \lambda = \lambda_0 + p^r \lambda_1
,\end{align*}
where \(\widehat{L}_r(\lambda_0)\) is a simple \(G_r{\hbox{-}}\)module
and \(\lambda_0 \in X_r(T)\).

\begin{proposition}[?]

For each \(\lambda\in X(T)\) and \(i\in {\mathbb{N}}\), there exists an
isomorphism of \(G{\hbox{-}}\)modules
\begin{align*}  \left\{{\substack{X(T)}}\right\} \ \widehat{L}_
H^i(\lambda) = R^i \mathop{\mathrm{ind}}_{G_r B}^G \widehat{Z}_r'( \lambda  )
.\end{align*}

\end{proposition}

\begin{proof}[?]

We can compose the two functors to get a Grothendieck-type spectral
sequence
\begin{align*}  
E_2^{m, n} = R^m \mathop{\mathrm{ind}}_{G_r B}^{G} \qty{ R^n \mathop{\mathrm{ind}}_{B}^{G_r B}(\lambda) }
\Rightarrow
R^{m+n} \mathop{\mathrm{ind}}_B^G(\lambda)
,\end{align*}
which follows from induction being transitive. Note that
\(\mathop{\mathrm{ind}}_{B}^{G_r B}({-})\) is exact, since coinduction
is given by
\(\operatorname{dist}(G_rB)\otimes_{\operatorname{Dist}(B)}\lambda \cong \operatorname{Dist}(U_r^+)\otimes_k \lambda\)
is tensoring over a field, and this is dual to induction. Thus
\(R^{>0} \mathop{\mathrm{ind}}_B^{G_r B}(\lambda) = 0\) and the spectral
sequence collapses to yield
\begin{align*}  
R^m \mathop{\mathrm{ind}}_{G_r B}^G R^0 \mathop{\mathrm{ind}}_{B}^{G_r B} 
= R^m \mathop{\mathrm{ind}}_{G_r B}^G \mathop{\mathrm{ind}}_{B}^{G_r B} 
= R^m \mathop{\mathrm{ind}}_B^G(\lambda)
,\end{align*}
where we can just note that
\(\mathop{\mathrm{ind}}_B^{G_r B}(\lambda) = \widehat{Z}_r'(\lambda)\).

\end{proof}

Recall \emph{Kempf's vanishing theorem}: if \(\lambda \in X(T)_+\) is a
dominant weight, then \(H^{>0}(\lambda) = 0\).

\begin{definition}[$p\dash$filtration, due to Steve Donkin]

Let \(M\in {\mathsf{G}{\hbox{-}}\mathsf{Mod}}\), then \(M\) has a (good)
\textbf{\((p, r){\hbox{-}}\)filtration} iff there exists a sequence of
\(G{\hbox{-}}\)modules
\begin{align*}  
0 = M_0 \subseteq M_1 \subseteq M_2 \subseteq \cdots \subseteq M_s = M
\end{align*}
such that
\(M_i / M_{i+1} \cong L(\lambda_0) \otimes H^0(\lambda_1)^{(r)}\) where
\(\lambda_0 \in X_r(T)\) (so the first time is irreducible) and
\(\lambda_1 \in X(T)_+\), so the second term is twisted.

\end{definition}

\begin{remark}

Question due to Jantzen: let \(\lambda \in X(T)_+\). Does
\(H^0(\lambda)\) have a good \((p, r){\hbox{-}}\)filtration?

\end{remark}

This question was open for a while, until the following was found:

\begin{proposition}[Parshall-Scott, 2013]

If \(p\geq 2(h-1)\) and Lusztig's character formula holds for \(G\),
then \(H^0(\lambda)\) has a good \((p, r)\) filtration.

\end{proposition}

\begin{proposition}[Bendell-Nakano-Pillen-Sobaje, 2019]

There are counterexamples to Jantzen's question. Example: \(\Phi = G_2\)
and \(p=2\).

\end{proposition}

\begin{quote}
Later: we'll see how to construct these filtrations by factoring
induction into intermediate inductions.
\end{quote}

\begin{theorem}[?]

Let \(\lambda \in X(T)_+\) and assume every composition factor of the
baby Verma \(\widehat{Z}_r'(\lambda)\) has the form
\(\widehat{L}_r(\mu_0 + p^r \mu_1) = \widehat{L}_r(\mu_0) \otimes p^r \mu_1\)
where \(\mu_0\in X_r(T)\) and \(\mu_1\in X(T)\) is any weight. Suppose
further that
\({\left\langle {\mu_1 + \rho},~{\beta {}^{ \vee }} \right\rangle}\geq 0\)
for all \(\beta\in \Delta\) (so it's ``pretty dominant''). Then
\(H^0(\lambda)\) has a good \((p, r)\) filtration, and moreover
\begin{align*}  
[ \widehat{Z}_r'(\lambda) : \widehat{L}_r(\mu_0) \otimes p^r \mu_1 ]
= [ H^0(\lambda) : L(\mu_0) \otimes H^0(\mu_1)^{(r)} ]
.\end{align*}

\end{theorem}

\begin{proof}[?]

Suppose \(\widehat{L}_r(\mu_0 + p^r\mu_1)\) is a composition factor of
\(\widehat{Z}_r'\). Then since we have \(G{\hbox{-}}\)modules, we can
use the tensor identity to write
\begin{align*}  
R^i \mathop{\mathrm{ind}}_{G_r B}^{G} L_r(\mu_0) \otimes p^r \mu_1
&= L_r(\mu_0) \otimes R^i \mathop{\mathrm{ind}}_{G_r B}^{G} p^r \mu_1 \\
&= L_r(\mu_0) \otimes H^i(\mu_1)^{(r)}
,\end{align*}
where the last equality follows from a theorem we won't prove here. We
can set \(i=0\) to yield
\begin{align*}  
\mathop{\mathrm{ind}}_{G_r B}^{G} L_r(\mu_0) \otimes p^r \mu_1
\cong L_r(\mu_0) \otimes H^0(\mu_1)^{(r)}
.\end{align*}
Recall that
\(H^0(\lambda) = \mathop{\mathrm{ind}}_{G_r B}^{G} \widehat{Z}_r'(\lambda)\),
so we'll take a composition series for \(\widehat{Z}_r'(\lambda)\) and
apply the induction functor to it. So let such a composition series be
given by
\begin{align*}  
0\subseteq N_0 \subseteq N_1 \subseteq \cdots \subseteq N_s = \widehat{Z}_r'(\lambda)
,\end{align*}
where \(N_i / N_{i-1} \cong L(\mu_0) \otimes p^r \mu_1\) for some
\(\mu_0\in X_r(T)\) and \(\mu_1\in X(T)\). Now apply the functor
\(\mathop{\mathrm{ind}}_{G_r B}^{G}({-})\) which yields
\begin{align*}  
0\subseteq \cdots \subset \mathop{\mathrm{ind}}_{G_r B}^{G} N_i \subseteq \cdots \subseteq H^0(\lambda)
.\end{align*}

Question: is this a good \((p, r)\) filtration?

\begin{warnings}

Note that if we have
\begin{align*}  
0 \to N_1 \to N_2 \to N_2/N_1 \to 0
\end{align*}
this yields
\begin{align*}  
0 \to \mathop{\mathrm{ind}}N_1 \to \mathop{\mathrm{ind}}N_2 \to \mathop{\mathrm{ind}}(N_2/N_1) \to R^1 \mathop{\mathrm{ind}}N_1 \to \cdots
.\end{align*}

Here we need
\(\mathop{\mathrm{ind}}(N_2/N_1) \cong \mathop{\mathrm{ind}}N_2 / \mathop{\mathrm{ind}}N_1\),
so we need to show \(R^1 \mathop{\mathrm{ind}}N_1 = 0\).

\end{warnings}

Using the tensor identity we can write
\begin{align*}  
R^1 \mathop{\mathrm{ind}}N_1
&= R^1 \mathop{\mathrm{ind}}_{G_r B}^G L_r(\sigma_0) \otimes p^r \sigma_0 \\
&= L_r(\sigma_0) \otimes\qty{R^1 \mathop{\mathrm{ind}}_{G_r B}^G \sigma_1 }^{(r)}
\end{align*}
and
\({\left\langle {\sigma_1 + \rho},~{\beta {}^{ \vee }} \right\rangle} \geq 0\),
so \(R^1 \mathop{\mathrm{ind}}_{G_r B}^G \sigma_1 = 0\). Thus we can
extend the region from Kempf's vanishing slightly:

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-06-14-38-20.png}
\caption{Image}
\end{figure}

\end{proof}

Finding composition factors for the \(\widehat{Z}_r'\) is in general a
hard problem: if we had this, we'd have the characters of the
irreducibles. Some combinatorics can be used here.

\hypertarget{strong-linkage-1}{%
\subsection{Strong Linkage}\label{strong-linkage-1}}

Note that strong linkage for \(H^0(\lambda)\) implies strong linkage for
\(\widehat{Z}_r'(\lambda)\).

\begin{theorem}[?]

Let \(\lambda, \mu \in X(T)\), then if
\([\widehat{Z}_r'(\lambda): \widehat{L}(\mu)] \neq 0\), then
\(\mu\uparrow\lambda\) and \(\mu\in W_p\cdot\lambda + p^r X(T)\).

\end{theorem}

\begin{proof}[?]

Note that \(\widehat{Z}_r'(\lambda)\) is finite dimensional. Idea:
tensor by a 1d rep to make all composition factors dominant. Then for
any weight \(\lambda\), we can find a large enough weight that moves
\(\lambda\) into the dominant chamber:

\includegraphics{figures/image_2020-11-06-14-45-02.png}

I.e., we can tensor by \(p^r v\) for \(v\) large so that
\(\widehat{Z}_r'(\lambda) \otimes p^r v\) has composition factors if the
form \(L(\mu_0) \otimes p^r \mu_1\) with
\({\left\langle {\mu_1 + \rho},~{\beta {}^{ \vee }} \right\rangle} \geq 0\)
for all \(\beta\in\Delta\).

Then \(\mu + p^r v \uparrow \lambda p^r v\), which implies
\(\mu\uparrow \lambda\), and so using strong linkage we have a
\(p{\hbox{-}}\)filtration on \(H^0(\lambda + p^r v)\).

\end{proof}

Next time: extensions in \({\mathsf{G_r T}{\hbox{-}}\mathsf{Mod}}\) and
the Steinberg module (very important in representation theory, has some
nice properties).

\hypertarget{monday-november-09}{%
\section{Monday, November 09}\label{monday-november-09}}

\hypertarget{strong-linkage-2}{%
\subsection{Strong Linkage}\label{strong-linkage-2}}

We have two categories:

\begin{itemize}
\tightlist
\item
  \(G_r T\), with a notion of \emph{strong linkage}, and
\item
  \(G_r\), which instead only has \emph{linkage}.
\end{itemize}

We'll restate a few theorems.

\begin{theorem}[?]

\envlist

Let \(\lambda, \mu \in X(T)\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  If \([\widehat{Z}_r(\lambda) : \widehat{L}_r(\mu) ]_{G_r T} \neq 0\),
  then \(\mu \uparrow \lambda\) are strongly linked.
\item
  If \([{Z}_r(\lambda) : {L}_r(\mu) ]_{G_r} \neq 0\), then
  \(\mu \in W_p \cdot\lambda + p^r X(T)\).
\end{enumerate}

\end{theorem}

\begin{example}[?]

In the case of \(\Phi = A_2\), we'll consider the two different
categories.

We have the following picture for \(\widehat{Z}\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-09-14-02-01.png}
\caption{Image}
\end{figure}

Considering \(X_1(T)\) and
\([\widehat{Z}_1(\lambda) : \widehat{L}_1(\mu)] \neq 0\), and
\(\widehat{Z}_1(\lambda)\) has 6 composition factors as
\(G_1T{\hbox{-}}\)modules.

On the other hand, for \(Z\), we have the following:

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-09-14-05-34.png}
\caption{Image}
\end{figure}

This again has 6 composition factors, obtained by ??

\todo[inline]{What's the main difference?}

\end{example}

\hypertarget{extensions}{%
\subsection{Extensions}\label{extensions}}

Let \(\lambda, \mu \in X(T)\). We can use the Chevalley
anti-automorphism (essentially the transpose) to obtain a form of
duality for extensions:
\begin{align*}  
\operatorname{Ext} _{G_r T}^j \qty{ \widehat{L}_r(\lambda), \widehat{L}_r(\mu) } 
= 
\operatorname{Ext} _{G_r}^j \qty{ \widehat{L}_r(\mu), \widehat{L}_r(\lambda) } \qquad \text{for }j\geq 0
.\end{align*}

We have a form of a weight space decomposition
\begin{align*}  
\operatorname{Ext} _{G_r}^j \qty{L_r(\lambda), L_r(\mu) }
= \bigoplus_{\gamma \in X(T)} \operatorname{Ext} _{G_r}^j \qty{L_r(\lambda), L_r(\mu) }_{\gamma}
\end{align*}
where we are taking the fixed points under the torus \(T\) action on the
first factor (for which \(T_r\) acts trivially). We can write this as
\begin{align*}  
\cdots 
&= \bigoplus_{\gamma \in X(T)} \operatorname{Ext} _{G_r}^j \qty{L_r(\lambda), L_r(\mu) \otimes\gamma } \\
&= \bigoplus_{\gamma \in X(T)} \operatorname{Ext} _{G_rT}^j \qty{L_r(\lambda), L_r(\mu) \otimes p^r v } \\
&= \bigoplus_{v \in X(T)} \operatorname{Ext} _{G_rT}^j \qty{ \widehat{L}_r(\lambda), \widehat{L}_r(\mu + p^r v) }
.\end{align*}

So if we know extensions in the \(G_r\) category, we know them in the
\(G_r T\) category.

There is an isomorphism
\begin{align*}  
\operatorname{Ext} _{G_r T}^1 \qty{ \widehat{L}_r(\lambda), \widehat{L}_r(\mu) } 
\cong \mathop{\mathrm{Hom}}_{G_R T}\qty{ \mathop{\mathrm{Rad}}_{G_r T} \widehat{Z}_r(\lambda), \widehat{L}_r(\mu) }
.\end{align*}

Finally, for \(\lambda, \mu \in X(T)\), if the above
\(\operatorname{Ext} ^1\) vanishes, then \(\lambda \in W_p \cdot \mu\)
(i.e.~\(\lambda\) and \(\mu\) are linked).

\hypertarget{the-steinberg-modules}{%
\subsection{The Steinberg Modules}\label{the-steinberg-modules}}

\begin{example}[Steinberg]

Consider \(A_2\):

\begin{figure}
\centering
\includegraphics{figures/image_2020-11-09-14-16-57.png}
\caption{Image}
\end{figure}

Taking the representation corresponding to \((p-1, p-1)\) yields the
``first Steinberg module''
\begin{align*}  
L(p-1, p-1) = L((p-1)\rho) \coloneqq\operatorname{St}_1 
.\end{align*}

In this case, we have an equality of many modules:

\begin{align*}  
H^0((p-1) \rho) =
L((p-1) \rho) =
V((p-1) \rho) =
T((p-1) \rho)
.\end{align*}

\end{example}

\begin{definition}[Steinberg Modules]

The \(r\)th \textbf{Steinberg module} is defined to be
\(L((p^r-1)\rho)\).

\end{definition}

\begin{remark}

In general, we have
\begin{align*}  
L((p^r-1)\rho) = 
H^0((p^r-1)\rho) = 
V((p^r-1)\rho)
.\end{align*}

We also have
\begin{align*}  
\widehat{Z}_r((p^r-1)\rho) \cong
L((p^r-1)\rho) \downarrow_{G_r T}
.\end{align*}

\end{remark}

\begin{theorem}[?]

The Steinberg module is both injective and projective as both a
\(G_r{\hbox{-}}\)module and a \(G_rT{\hbox{-}}\)module.

\end{theorem}

\begin{proof}[?]

It suffices to prove that \({\mathsf{Stk}}_r\) is projective over
\(G_r T\), then by a previous theorem, it will also be projective over
\(G_r\). Let \(\widehat{L}_r(\mu)\) be a simple
\(G_rT{\hbox{-}}\)module, and consider
\begin{align*}  
\operatorname{Ext} _{G_rT}^1({\mathsf{Stk}}_r, \widehat{L}_r(\mu)) =
\operatorname{Ext} _{G_rT}^1(\widehat{L}_r((p^r-1)\rho), \widehat{L}_r(\mu))
.\end{align*}
If we show this is zero for every simple module, the result will follow.

Suppose \((p^r-1)\rho\not< \mu\). In this case, the RHS above is zero.

\todo[inline]{Missed why: something to do with radical of the first term?}

Otherwise, we have
\begin{align*}  
\operatorname{Ext} _{G_rT}^1(\widehat{L}_r(\mu), {\mathsf{Stk}}_r) =
\mathop{\mathrm{Hom}}_{G_rT}(\mathop{\mathrm{Rad}}(\widehat{Z}_r(\mu)) , {\mathsf{Stk}}_r)
.\end{align*}
Suppose that the RHS is nonzero. Then
\(\mathop{\mathrm{Rad}}\qty{\widehat{Z}_r(\mu)} \twoheadrightarrow{\mathsf{Stk}}_r\),
and thus
\begin{align*}
\dim \mathop{\mathrm{Rad}}\qty{ \widehat{Z}_r(\mu) } \geq \dim {\mathsf{Stk}}_r = p^{r{\left\lvert {\Phi^+} \right\rvert}}
\end{align*}
But we know that
\begin{align*}
\dim \mathop{\mathrm{Rad}}\qty{\widehat{Z}_r(\mu)} < \dim \widehat{Z}_r(\mu) = p^{r{\left\lvert {\Phi^+} \right\rvert}}
,\end{align*}
so we've reached a contradiction and the hom must be zero.

\end{proof}

\begin{proposition}[Open Conjecture, Donkin, MSRI 1990: 'DFilt Conjecture']

Let \(G\) be a reductive group and \(M\) a finite-dimensional
\(G{\hbox{-}}\)module. Then \(M\) has a good
\((p, r){\hbox{-}}\)filtration iff \({\mathsf{Stk}}_r \otimes M\) has a
good filtration.

\end{proposition}

\begin{remark}

See NK (Nakano-Kildetoft, 2015) and BNPS (Bendel-Nakano-Pillen-Subaje,
\(2018{\hbox{-}}\)).

\end{remark}

\begin{remark}[Important! What we've been working toward stating]

The forward direction is equivalent to the statement that
\({\mathsf{Stk}}_r \otimes L(\lambda)\) has a good filtration for
\(\lambda \in X_r(T)\).

\end{remark}

\begin{proposition}[Conjecture]

The Dfilt conjecture in the forward direction holds for all \(p\).

\end{proposition}

\begin{remark}

This is known for \(p\geq 2h-4\)? BNPS has shown that this holds for all
rank 2 groups, which is strong evidence. The reverse implication is
\textbf{not} true: BNPS-Crelle 2020 shows that for \(\Phi = G_2, p=2\),
there exists an \(H^0(\lambda)\) that does not have a good
\((p, r){\hbox{-}}\)filtration. There is a similar conjecture for
tilting modules (``DTilt'').

\end{remark}

Main difference to category \({\mathcal{O}}\): infinitely many highest
weight representations?

Upcoming:

\begin{itemize}
\tightlist
\item
  Viewing the \(G_r T\) category as ``almost'' a highest weight category
\item
  Defining standard and costandard modules \(\Delta(\lambda)\) and
  \(\nabla(\lambda)\).
\item
  Injective \(G_r T{\hbox{-}}\)modules
\item
  Results of Verma and Humphreys on the Lifting Conjecture: does every
  \(G_r T{\hbox{-}}\)module come from a \(G{\hbox{-}}\)module?
\item
  Donkin's Tilting Module Conjecture (DTilt)
\end{itemize}

\begin{proposition}[Statement of DTilt Conjecture, Kildetoft-Nakano-Subaje MSRI 1990]

DFilt implies DTilt, and DTilt implies the forward direction of DFilt.

\end{proposition}

\hypertarget{friday-november-13}{%
\section{Friday, November 13}\label{friday-november-13}}

\hypertarget{review-3}{%
\subsection{Review}\label{review-3}}

Review: we're considering \(G_r T{\hbox{-}}\)modules, with several
associated modules of interest:

\begin{itemize}
\tightlist
\item
  Simple modules \(\widehat{L}_r(\lambda)\) for \(\lambda \in X(T)\)
\item
  Intermediate modules \(\nabla(\lambda) = \widehat{Z}_r'(\lambda)\) and
  \(\Delta(\lambda) = \widehat{Z}_r(\lambda)\).
\item
  Injective and projective modules \(\widehat{Q}_r(\lambda)\)
\end{itemize}

\begin{theorem}[?]

Let \(M\) be a \(G_r T{\hbox{-}}\)module of finite dimension. Then \(M\)
has a \(\widehat{Z}_r\) filtration \(\iff\) \(M\downarrow_{B_r}\) is
projective.

\end{theorem}

\begin{remark}

From this, the multiplicity \([M: \widehat{Z}_r(\mu)]\) (the number of
times \(\widehat{Z}_r(\mu)\) appears in a \(\widehat{Z}_r\) filtration )
is well-defined. Moreover, we have a decomposition
\begin{align*}  
M\downarrow_{B_r} = \bigoplus_{\mu} Z_r(\mu)\downarrow_{B_r}
,\end{align*}
where the sum contains as many terms as the number of factors that
appear. We have \(Z_r(\mu)\downarrow_{B_r} \twoheadrightarrow\mu\),
making is the projective cover of \(\mu\) and thus indecomposable. We
can then apply the Krull-Schmidt theorem.

\end{remark}

\hypertarget{reciprocity}{%
\subsection{Reciprocity}\label{reciprocity}}

Consider \(\widehat{Q}_r(\lambda)\), a projective
\(G_r T{\hbox{-}}\)module. Note that it also happens to be injective. We
saw before that the functor
\(\mathop{\mathrm{coInd}}_{B_r T}^{G_r T}({-})\) is exact, and thus
\(\widehat{Q}_r(\lambda)\downarrow_{B_r T}\) being projective implies
that \(\widehat{Q}_r(\lambda)\downarrow{B_r}\) is also projective. This
implies that \(\widehat{Q}_r(\lambda)\) has a
\(\widehat{Z}_r{\hbox{-}}\)filtration.

Thus the multiplicity can be computed as
\begin{align*}
[\widehat{Q}_r(\lambda) : \widehat{Z}_r(\mu)] 
&= [\widehat{Q}_r \downarrow{B_r T} : \widehat{Z}_r(\mu)] \\
&= \dim \mathop{\mathrm{Hom}}_{B_r T}\qty{\widehat{Q}_r(\lambda), \mu } \\
&= \dim \mathop{\mathrm{Hom}}_{B_r T}\qty{ \widehat{Q}_r(\lambda), \mathop{\mathrm{ind}}_{B_r T}^{G_r T} \mu } && \text{by Frobenius reciprocity} \\
\end{align*}

\begin{exercise}[?]

Show that
\begin{align*}  
[M: S] = \dim \mathop{\mathrm{Hom}}_A( P(S), \mu) = [\mathop{\mathrm{ind}}_{B_r T}^{B_r T} \mu : \widehat{L}_r(\lambda)]
.\end{align*}

\end{exercise}

We can thus continue this computation as
\begin{align*}  
\cdots 
&= [\widehat{Z}_r'(\mu) : \widehat{L}_r(\mu)] \\
&= [\widehat{Z}_r(\mu) : \widehat{L}_r(\mu)]
,\end{align*}
since
\(\operatorname{ch}\widehat{Z}_r (\mu) = \operatorname{ch}\widehat{Z}_r'(\mu)\).

Thus we have the following reciprocity theorem

\begin{theorem}[Humphreys]

\begin{align*}  
[\widehat{Q}_r(\lambda): \widehat{Z}_r(\mu)] = [\widehat{Z}_r(\mu) : \widehat{L}_r (\lambda) ]
.\end{align*}

\end{theorem}

\begin{remark}

This is hard to prove in the \(G_r\) category, need to work in the
\(G_r T\) category and descend. However, this reciprocity does also work
for \(G_r\).

\end{remark}

\begin{example}[?]

For \(G = {\operatorname{SL}}_2\), consider \(G_1 T\) or \(G_1\) where
\(\lambda = 0,1,2,\cdots, (p-1)\). We have a notion of \emph{linkage}:
\(\lambda, \mu\) are in the same \(G_1\) block iff
\(\lambda + \mu = p-2\). Note that \(\lambda = p-1\) is in its own
block.

We have
\begin{align*}  
Z_r(\lambda) = \mathop{\mathrm{coInd}}_{B_1^+}^{G_1} \lambda \twoheadrightarrow L(\lambda)
.\end{align*}
If \(\lambda + \mu = p-2\), then we have the following situation:

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-11-13-14-15-25.png}
\caption{Image}
\end{figure}

Taking \(\lambda = p-1\), we have
\(Z_r((p-1)\rho) = L(p-1) = {\mathsf{Stk}}_1\).

Applying reciprocity, we gave
\begin{align*}  
[Q_1(0) : Q_1(\mu)] = [Q_1(\mu): L(0)]
.\end{align*}
Since \(Q_1(0)\) has factors \(Z_1(0)\) and \(Z_1(p-2)\), we have

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-11-13-14-18-03.png}
\caption{Image}
\end{figure}

We can identify the two filtrations here:

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-11-13-14-19-49.png}
\caption{Image}
\end{figure}

Similarly, for \(Q_1(p-2)\) we have

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-11-13-14-21-17.png}
\caption{Image}
\end{figure}

We have

\begin{itemize}
\item
  \(\dim \widehat{Q}_1(\lambda) = 2p\) for \(\lambda \neq p-1\)
\item
  \(\dim \widehat{Q}_1(p-1) = p\) for \(\lambda = p-1\).
\end{itemize}

\end{example}

\begin{remark}[Some historical background on reciprocity laws]

Some work predated the BGG Category \({\mathcal{O}}\). For finite
groups, a notion of CDE triangles was worked out.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Pollack (1967) computed the structure of projectives for \(G_1\) in
  \(G = {\operatorname{SL}}_2\).
\item
  Humphreys (1971) proved reciprocity for \(G_1\). (They were students
  together.)
\item
  Bernstein-Gelfand-Gelfand (1976): developed machinery for Category
  \({\mathcal{O}}\), crediting Humphreys.
\item
  Roche-Caridi (1980): Proved reciprocity for generalized Verma modules.
\item
  BGG Algebra, Irving: A more axiomatic approach.
\item
  CPS (1988): Generalized to highest weight categories, also attributed
  to Humphreys.
\item
  Holmes-Nakano (1987): Proved when there is a triangular decomposition
  \(A = A^- A_0 A^+\), looked at filtrations and reciprocity, applies to
  Lie algebras of Cartan type.\footnote{Simple Lie algebras in
    characteristic \(p\) with a triangular decomposition which is highly
    non-symmetric (negative part is typically smaller).}
\end{enumerate}

\end{remark}

\hypertarget{toward-lifting-conjectures}{%
\subsection{Toward Lifting
Conjectures}\label{toward-lifting-conjectures}}

Recall that \(G_r T \subseteq G\).

\textbf{Question}: Given \(\widehat{Q}_r(\lambda)\) for a restricted
weight \(\lambda \in X_r(T)\), does \(\widehat{Q}_r(\lambda)\)
\emph{lift} to \(G\)? I.e., does there exist a \(G{\hbox{-}}\)module
\(M(\lambda)\) such that
\(M(\lambda)\downarrow_{G_r T} = \widehat{Q}_r(\lambda)\)?

\begin{remark}

Note that \(L_r(\lambda)\) for \(\lambda\in X_r(T)\) lifts to \(G\),
since \(L(\lambda)\downarrow_{G_r T} = \widehat{L}_r(\lambda)\).

\end{remark}

\begin{theorem}[?]

Let \(p > 2h-2\) and \(\lambda \in X_r(T)\), then
\(\widehat{Q}_r(\lambda)\) has a lift to a \(G\) structure.

\end{theorem}

\begin{remark}[Some history]

\envlist

\begin{itemize}
\item
  One can prove that the \(G\) structure is unique, since this turns out
  to be a projective module in an appropriate category (which we won't
  get into).
\item
  Ballard (1970s) proved the theorem for \(p>3h-3\).
\item
  Jantzen (late 1970s) lowered the bound to \(p>2h-2\)
\item
  Amazingly, no one has been able to lower this bound! This is currently
  an open question.
\item
  For \(G = {\operatorname{SL}}_2, {\operatorname{SL}}_3\), it is known
  that \(\widehat{Q}_r(\lambda)\) has a \(G\) structure for all \(p\).
\end{itemize}

\end{remark}

\hypertarget{donkins-tilting-module-conjecture}{%
\subsubsection{Donkin's Tilting Module
Conjecture}\label{donkins-tilting-module-conjecture}}

From MSRI, 1990. Some notation first: for \(\lambda \in X_r(T)\), define
\begin{align*}  
\widehat{\lambda} \coloneqq 2(p-1) \rho + w_0 \lambda
.\end{align*}

\begin{conjecture}[?]

Let \(G\) be a semisimple simply connected algebraic group over
\(k = \mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern 1.5mu_p\)
for some \(p\). Then
\begin{align*}  
T(\widehat{\lambda}) \downarrow_{G_r T} \cong \widehat{Q}_r(\lambda)
.\end{align*}

\end{conjecture}

\todo[inline]{Something about DTilt conjecture being true for $p>2h-2$.}

Next time:

\begin{itemize}
\item
  Proof of theorem
\item
  \(\widehat{Q}_r(\lambda) \divides {\mathsf{Stk}}_r \otimes L(\sigma)\)
  as \(G{\hbox{-}}\)modules, and is also projective as a
  \(G_r T{\hbox{-}}\)module.
\item
  Find a \(G{\hbox{-}}\)summand \(M(\lambda)\) such that
  \(M(\lambda)\downarrow_{G_r T} = \widehat{Q}_r (\lambda)\).
\item
  More with injective modules.
\item
  Possibly something about cohomology of Frobenius kernels.
\end{itemize}

\hypertarget{monday-november-16}{%
\section{Monday, November 16}\label{monday-november-16}}

We'll focus on the following theorem:

\begin{theorem}[Special case of Humphreys-Verma conjecture, 1973]

Let \(p> 2h-2\) and \(\lambda\in X_r(T)\), then
\(\widehat{Q}_r(\lambda)\) has a \(G{\hbox{-}}\)structure.

\end{theorem}

Recall Donkin's tilting conjecture:

\begin{conjecture}[DTilt]

Let \(\lambda\in X_r(T)\), then
\begin{align*}  
T(\widehat{\lambda})\downarrow_{G_r T} = \widehat{Q}_r(\lambda)
.\end{align*}
where \(\lambda = 2(p-1)\rho + w_0\lambda\).

\end{conjecture}

\begin{remark}

There is a counterexample to DTilt conjecture due to
Bendel-Pillen-Nakano-Subaje (2019) to DTilt for \(\Phi = G_2\) and
\(p=2\). Note that this doesn't apply to the previous conjecture, which
could still be true.

\end{remark}

\hypertarget{existence-of-ghbox-structures-some-preliminaries}{%
\subsection{\texorpdfstring{Existence of \(G{\hbox{-}}\)Structures: Some
Preliminaries}{Existence of G\{\textbackslash hbox\{-\}\}Structures: Some Preliminaries}}\label{existence-of-ghbox-structures-some-preliminaries}}

We want to consider \(G{\hbox{-}}\)structures on injective modules. Let
\(V, W\) be \(G{\hbox{-}}\)modules, then
\begin{align*}  
\mathop{\mathrm{Hom}}_{G_r}(V, W) = [V {}^{ \vee }\otimes W]^{G_r}
,\end{align*}
which is a module on which \(G_r\) acts trivially where we pull back the
action using the map \(G \to G/G_r\). Moreover, there exists a
\(G{\hbox{-}}\)modules \(M\) such that
\(\mathop{\mathrm{Hom}}_{G_r}(V, W) = M^{(r)}\) twisted \(r\) times.

We can consider blocks, e.g.~by considering
\({\operatorname{pr}}_\nu M\). This is the sum of all submodules with
composition factors in the same block as \(L(\nu)\). We can write
\(M = \bigoplus_{\nu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}} {\operatorname{pr}}_\nu M\).

\begin{figure}
\centering
\includegraphics[width=3.64583in,height=\textheight]{figures/image_2020-11-16-14-03-33.png}
\caption{\(\mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\)}
\end{figure}

Thus we can write
\begin{align*}  
\mathop{\mathrm{Hom}}_{G_r}(V, M) = \bigoplus_{\nu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}} \mathop{\mathrm{Hom}}_{G_r}^\nu(V, W) = \qty{ {\operatorname{pr}}_\nu M }^{(r)}
.\end{align*}

Note that for \(p>h\) the Coxeter number,
\(0\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}\),
since
\begin{align*}  
0 \leq {\left\langle {0+\rho},~{\alpha {}^{ \vee }} \right\rangle} = {\left\langle {\rho},~{\alpha_0 {}^{ \vee }} \right\rangle} = h-1 < p
,\end{align*}
where
\(h = {\left\langle {\rho},~{\alpha_0 {}^{ \vee }} \right\rangle}\).

We can then write
\begin{align*}  
\mathop{\mathrm{Hom}}_G(V, W) = \mathop{\mathrm{Hom}}_{G_r}(V, W)^{G_r} \subseteq \mathop{\mathrm{Hom}}_{G_r}^0(V, W)
,\end{align*}
where the middle term involves trivial modules.

\hypertarget{sketch-of-proof}{%
\subsubsection{Sketch of Proof}\label{sketch-of-proof}}

\begin{claim}

Let \(\lambda \in X_r(T)\) and consider
\(M \coloneqq{\mathsf{Stk}}_r \otimes L( (p^r-1)\rho + w_0 \lambda )\).
Then \(\widehat{Q}_r(\lambda)\) is a direct summand of \(M\) as a
\(G_r T{\hbox{-}}\)module.

\end{claim}

In other words, if we restrict this down to \(G_r T{\hbox{-}}\)modules,
we get a \(G_r T{\hbox{-}}\)summand. Note that if DFilt holds, \(M\) is
a Tilting module.

Recall that \(M\) is a projective and injective \(G_r T{\hbox{-}}\)
module. It suffices to show that
\(\widehat{L}_r(\lambda)\in \mathop{\mathrm{Soc}}_{G_r T} M\), for which
we look at the hom space
\begin{align*}  
\mathop{\mathrm{Hom}}_{G_r T} (\widehat{L}_r(\lambda), {\mathsf{Stk}}_r \otimes L( (p^r-1)\rho + w_0 \lambda ) )
&= \mathop{\mathrm{Hom}}_{G_r T} (\widehat{L}_r(\lambda) \otimes{\mathsf{Stk}}_r, L( (p^r-1)\rho + w_0 \lambda ) ) \\
&= \mathop{\mathrm{Hom}}_{G_r T} ({\mathsf{Stk}}_r, \widehat{L}_r(-w_0 \lambda) \otimes L( (p^r-1)\rho + w_0 \lambda ) ) 
,\end{align*}

\todo[inline]{Something about $(p^r-1)\rho$ being a highest weight? And the last $L$ term being nonzero?}

Let \(Q_\lambda\) be the injective hull of \(L_\lambda\) as a
\(G{\hbox{-}}\)module. This yields an injection
\(L(\lambda)\hookrightarrow M\). Since we also have a map into the
injective hull, we can extend:

\begin{center}
\begin{tikzcd}
& Q_\lambda \\
L(\lambda) \ar[r, hook]\ar[ur, hook] & M\ar[u, dotted, "\exists \phi"] \supseteq \widehat{Q}_r(\lambda)
\end{tikzcd}
\end{center}

Moreover, \(\phi:\widehat{Q}_r(\lambda)\to Q_\lambda\) is an injective
map since \(L(\lambda) \subseteq \mathop{\mathrm{Soc}}_{G_r}(\lambda)\).
Thus the image
\(\phi\qty{\widehat{Q}_r(\lambda) } \cong \widehat{Q}_r(\lambda)\) and
is a \(G_r T\) summand of \(\phi(M)\) since \(\widehat{Q}_r(\lambda)\)
is an injective \(G_r T{\hbox{-}}\)module. We have a split exact
sequence
\begin{align*}  
0 \to \phi(\widehat{Q}_r(\lambda)) \hookrightarrow\phi(M) \to ? \to 0
.\end{align*}

The idea is now that \(\phi(M)\) is a \(G{\hbox{-}}\)module, so we'll
show \(\phi(M)\) is indecomposable as both a \(G\) and a \(G_r T\)
module. In this case, we'll have
\begin{align*}  
\phi(M) = \phi(\widehat{Q}_r(\lambda)) \cong \widehat{Q}_r(\lambda)
.\end{align*}

It suffices to prove that
\(\mathop{\mathrm{Soc}}_{G_r} \phi(M) = L_r(\lambda)\). Note that that
it suffices to show there's only one summand, since the socle can't be
decomposed further, which will yield irreducibility. We have
\begin{align*}  
\mathop{\mathrm{Soc}}_G \phi(M) = L(\lambda)
.\end{align*}

We also know that
\(\mathop{\mathrm{Soc}}_G \phi(M) \subseteq \mathop{\mathrm{Soc}}_{G_r} \phi(M)\).
But if \(L(\mu) \subseteq \mathop{\mathrm{Soc}}_{G_r} \phi(M)\), then it
is also contained in \(\mathop{\mathrm{Soc}}_G \phi(M)\). Hence if
\(\mathop{\mathrm{Soc}}_{G_r}\phi(M)\) is isotypic of type
\(L(\lambda)\), we have a decomposition
\begin{align*}  
\mathop{\mathrm{Soc}}_{G_r}\phi(M) 
&= \bigoplus_{\sigma\in X_r(T)} L(\sigma) \otimes\mathop{\mathrm{Hom}}_{G_r}(L(\sigma), \phi(M)) \\
&= L(\lambda) \otimes\mathop{\mathrm{Hom}}_{G_r}(L(\lambda), \phi(M)) \\
&= \bigoplus_{\nu\in \mkern 1.5mu\overline{\mkern-1.5muC\mkern-1.5mu}\mkern 1.5mu_{\mathbb{Z}}} L(\lambda) \otimes\mathop{\mathrm{Hom}}_{G_r}^\nu (L(\lambda), \phi(M))
,\end{align*}
where we've broken this up into blocks. Note that \(\phi(M)\) is an
indecomposable \(G{\hbox{-}}\)module, thus using linkage we can conclude
that there is only one summand. By the previous statement, we have
\begin{align*}  
k = \mathop{\mathrm{Hom}}_{G}(L(\lambda), \phi(M))
 \subseteq \mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M))
,\end{align*}
and hence

\begin{equation} \mathop{\mathrm{Soc}}_{G_r} \phi(M) = L(\lambda) \otimes\mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M)) .\end{equation}

\begin{remark}

Up until now, we've only used that \(p>h\). If we show that the hom
space in the last equality is just the trivial module \(k\), then we're
done. This would imply that
\(\mathop{\mathrm{Soc}}_{G_r} \phi(M) = L(\lambda)\), in which case
\(\phi(M)\) is an indecomposable \(G_r T{\hbox{-}}\)module and
\(\phi(M) = \widehat{Q}_r(\lambda)\).

\end{remark}

The critical equation to show:

\begin{equation} \mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M)) = k .\end{equation}

Suppose that
\(\mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M)) \neq k\). Then
there exists a \(\nu\in X(T)_+ \cap W_p \cdot 0\) with \(\nu\neq 0\)
such that \(L(\nu)^{(r)} = L(p^r \nu)\) is a composition factor of
\(\mathop{\mathrm{Hom}}_{G_r}^0 (L(\lambda), \phi(M))\). This would
imply that \(L(\lambda) \otimes L(\nu)^{(r)} = L(\lambda+ p^r\nu)\) is a
composition factor of
\({\mathsf{Stk}}_r \otimes H^0( (p^r-1)\rho + w_0 \lambda)\).

The idea now is to check the weights in this module either by Weyl's
character formula or weight estimates. The upshot: for \(p>2h-2\),
\(\lambda+ p^r \nu\) is a weight if \(\nu = 0\). \(\hfill\blacksquare\)

How might this bound be lowered? Maybe the condition of the hom space
being trivial is too strong, since being indecomposable isn't equivalent
to having a simple socle.

\hypertarget{steinberg-tensor-product-theorem-for-injective-g_r-thbox-modules}{%
\subsection{\texorpdfstring{Steinberg Tensor Product Theorem for
Injective
\(G_r T{\hbox{-}}\)Modules}{Steinberg Tensor Product Theorem for Injective G\_r T\{\textbackslash hbox\{-\}\}Modules}}\label{steinberg-tensor-product-theorem-for-injective-g_r-thbox-modules}}

\begin{proposition}[?]

Let \(\lambda_0 \in X_r(T)\) and \(\lambda_1 \in X(T)\). Then there
exists an isomorphism of \(G_r T{\hbox{-}}\)modules
\begin{align*}  
\widehat{Q}_{r+1}(\lambda_0 + p^r \lambda_1)\downarrow_{G_r T} 
\,\, \cong \widehat{Q}_r(\lambda_0) \otimes\widehat{Q}_1(\lambda_1)^{(r)}
,\end{align*}
where the last term is a \(G_1 T{\hbox{-}}\)module with \(G_r\) acting
trivially, which thus becomes a \(G_r T{\hbox{-}}\)module.

\end{proposition}

\begin{remark}

Comparing this with the situation where \(\lambda_1 \in X(T)_+\), we get
\begin{align*}  
L(\lambda_0 + p^r \lambda_1) 
\cong L(\lambda_0) \otimes L(\lambda_1)^{(r)}
,\end{align*}
which is an isomorphism of \(G{\hbox{-}}\)modules.

\end{remark}

Next time: we'll complete injective modules, and if we have time, we'll
talk about cohomology.

\hypertarget{wednesday-november-18}{%
\section{Wednesday, November 18}\label{wednesday-november-18}}

\hypertarget{steinberg-tensor-product-formula}{%
\subsection{Steinberg Tensor Product
Formula}\label{steinberg-tensor-product-formula}}

Recall the Steinberg tensor product formula: Let \(\lambda \in X(T_+)\)
with \(\operatorname{ch}(k) = p\), then write
\begin{align*}  
\lambda = \sum_{j=1}^s p^j \lambda_j
\end{align*}
with \(\lambda_j \in X_1(T)\). Then
\begin{align*}  
L(\lambda) = L(\lambda_0) \otimes\bigotimes_{j=1}^s L(\lambda_j)^{(j)}
.\end{align*}

The goal is to find a formula resembling this for \(G\) and
\(G_r{\hbox{-}}\)modules.

\begin{proposition}[?]

Let \(\lambda_0 \in X_r(T), \lambda_1 \in X(T)\). Then there exists an
isomorphism of \(G_r T{\hbox{-}}\)modules
\begin{align*}  
\widehat{Q}_{r+1} (\lambda_0 + p^r \lambda_1) = \widehat{Q}_r (\lambda_0) \otimes\widehat{Q}_1(\lambda_1)^{(r)}
,\end{align*}
and by inflation, \(\widehat{Q}_1(\lambda)^{(r)}\) is a
\(G_r T{\hbox{-}}\)module.

\end{proposition}

\begin{proof}[?]

We have a decomposition of \(G_r T{\hbox{-}}\)modules:
\begin{align*}  
\mathop{\mathrm{Soc}}_{G_r} \widehat{Q}_{r+1}(\lambda) = \bigoplus_{\mu\in X_r(T)} \widehat{L}_r(\mu) \otimes\mathop{\mathrm{Hom}}_{G_r}(\widehat{L}_r(\mu), \widehat{Q}_{r+1}(\lambda) )
.\end{align*}

We can write

\begin{align*}  
\mathop{\mathrm{Soc}}_{G_{r+1}} \widehat{Q}_{r+1}(\lambda) 
&= \widehat{L}_{r+1}(\lambda) \\
&= \widehat{L}_r(\lambda_0) \otimes\widehat{L}_1(\lambda_1)^{(r)}
.\end{align*}
And since there can only be one summand, we can combine these two to
write

\begin{align*}  
\mathop{\mathrm{Soc}}_{G_r} \widehat{Q}_{r+1}(\lambda) = \widehat{L}_r(\lambda_0) \otimes\mathop{\mathrm{Hom}}_{G_r}(\widehat{L}_r(\lambda_0), \widehat{Q}_{r+1}(\lambda) )
,\end{align*}
which implies that
\begin{align*}  
\mathop{\mathrm{Soc}}_{G_{r+1} T} \hom_{G_r}(\widehat{L}_r(\lambda_0), \widehat{Q}_{r+1}(\lambda)  ) &\cong \widehat{L}_1(\lambda_1)^{(r)}
.\end{align*}

We want to show that this hom is injective, so we'll use the
Lydon-Hochschild-Serre spectral sequence
\begin{align*}  
E_2^{i, j} = \operatorname{Ext} _{G_{r+1} T/G_r}^i (N, \operatorname{Ext} ^j( \widehat{L}_r(\lambda_0), \widehat{Q}_{r+1}(\lambda) )  )
\Rightarrow
\operatorname{Ext} ^{i+j}_{G_{r+1} T} ( N\otimes\widehat{L}_r(\lambda_0), \widehat{Q}_{r+1}(\lambda) )
.\end{align*}

There will be a lot of vanishing here. We know that

\begin{itemize}
\tightlist
\item
  \(\widehat{Q}_{r+1}(\lambda)\) is an injective
  \(G_{r+1}T{\hbox{-}}\)module, and
\item
  \(\widehat{Q}_{r+1}(\lambda)\) is an injective
  \(G_{r}{\hbox{-}}\)module,
\end{itemize}

so \(E_2^{>0, 0} = 0\):

\begin{align*}  
\operatorname{Ext} _{G_{r}T/G_r}^i(N, \mathop{\mathrm{Hom}}_{G_r}( \widehat{L}_r(\lambda_0), \widehat{Q}_{r+1}(\lambda) )  )
.\end{align*}

We can thus conclude that
\(\hom_{G_r}( \widehat{L}_r(\lambda_0), \widehat{Q}_{r+1}(\lambda) )\)
is an injective \(G_{r+1}T/G_r{\hbox{-}}\)module. Thus
\begin{align*}  
\mathop{\mathrm{Hom}}_{G_r}(\widehat{L}_r(\lambda_0), \widehat{Q}_{r+1}(\lambda) ) \cong \widehat{Q}_{1}(\lambda_1)^{(r)}
.\end{align*}

To finish the proof, note that
\(\widehat{Q}_{r+1}(\lambda) \downarrow_{G_r T} = \bigoplus_{j=1}^m \widehat{Q}_r(\mu_j)\),
and thus
\begin{align*}  
\widehat{Q}_{r+1}(\lambda) = \bigoplus_{j-1}^m \widehat{Q}_r(\lambda_0) \otimes p^r v_j
.\end{align*}

Furthermore, at the level of characters we have
\(\sum e^{v_j} = \operatorname{ch}\widehat{Q}_1(\lambda_1)\) and thus
\begin{align*}  
\widehat{Q}_1(\lambda_1)^{(r)} = \bigoplus_{j=1}^m p^r v_j
.\end{align*}

\end{proof}

\begin{remark}

Suppose \(\lambda_0, \lambda_1, \lambda_{r-1}\in X(T)\), where we can
write \(\lambda = \sum_{i=0}^{r-1} \lambda_i p^i\). Then for characters
we have
\begin{align*}  
\widehat{Q}_{r}(\lambda) = \prod_{j=0}^{r-1} \operatorname{ch}\widehat{Q}_1(\lambda_j)^{(j)}
.\end{align*}

\end{remark}

Let \(L(\lambda)\) be an irreducible representation for \(G\), and set
\(Q_\lambda\) to be the injective hull of \(L(\lambda)\), so we have an
exact sequence \(0 \to L(\lambda) \hookrightarrow Q(\lambda)\).

\begin{proposition}[?]

Let \(\lambda\in X_r(T)\), and assume that the \(G_r T{\hbox{-}}\)
structure on \(\widehat{Q}_r(\lambda)\) can be lifted to \(G\). Then

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\lambda'\in X(T)_+\), then there exists an isomorphism of
  \(G{\hbox{-}}\)modules:
  \begin{align*}  
  \widehat{Q}_r(\lambda) \otimes Q_{\lambda'}^{(r)} \cong Q_{\lambda + p^r \lambda'}
  .\end{align*}
\item
  If \(\lambda'\in X(T)\) and \(r'>r\), then there exists an isomorphism
  of \(G_{r'} T{\hbox{-}}\)modules
  \begin{align*}  
  \widehat{Q}_r(\lambda) \otimes\widehat{Q}_{r' - r}(\lambda')^{(r)} \cong \widehat{Q}_{r'}(\lambda + p^r \lambda')
  .\end{align*}
\end{enumerate}

\end{proposition}

\begin{proof}[of a]

We want to show that
\(\widehat{Q}_r(\lambda) \otimes Q_{\lambda'}^{(r)}\) is an injective
\(G{\hbox{-}}\)module and that the socle of the LHS is equal to
\(L(\lambda + p^r \lambda')\).

\begin{claim}

Let \(N{~\trianglelefteq~}G\) be groups, and suppose
\(V_1 \in {\mathsf{G}{\hbox{-}}\mathsf{Mod}}\) is injective as an
\(N{\hbox{-}}\)module and \(V_2\) is injective as a
\(G/N{\hbox{-}}\)module. Then \(V_1 \otimes V_2\) is injective as a
\(G{\hbox{-}}\)module.

\end{claim}

\begin{proof}[of claim]

We'll again use the LHS spectral sequence,
\begin{align*}  
E_{2}^{i, j} = \operatorname{Ext} _{G/N}^i(k, \operatorname{Ext} _N^j(E, V_1\otimes V_2) )
\Rightarrow
\operatorname{Ext} _G^{i+j}(E, V_1 \otimes V_2)
.\end{align*}
Using the assumption that \(V_1\) is injective as an
\(N{\hbox{-}}\)module, we have collapsing
\begin{align*}  
\operatorname{Ext} _N^{>0}(E, V_1\otimes V_2) = \operatorname{Ext} _N^{>0} (E\otimes V_2, V_1)
.\end{align*}
and thus
\begin{align*}  
\operatorname{Ext} _{G/N}^{>0}(k, \mathop{\mathrm{Hom}}_N(E, V_1\otimes V_2) )
= \operatorname{Ext} _G^{>0}(N, V_1\otimes V_2)
= 0
.\end{align*}
For the first line, we can write \(\hom_N(E, V_1) \otimes V_2\). This
vanishes for every possible \(E\), making \(V_1\otimes V_2\) injective.

\end{proof}

In our situation, we'll be taking
\(N\coloneqq G_r {~\trianglelefteq~}G\), where \(G/G_r = G^{(r)}\). Thus
the claim proves part (a).

\end{proof}

\begin{proof}[of b]

Recall that the \(G{\hbox{-}}\)socle is contained in
\(\mathop{\mathrm{Soc}}_{G_r}\), yielding
\begin{align*}  
\mathop{\mathrm{Soc}}_{G_r} \widehat{Q}_r(\lambda) \otimes Q_{\lambda'}^{(r)} 
= \widehat{L}_R(\mu) \otimes\mathop{\mathrm{Hom}}_{G_r}( \widehat{L}_r(\mu), \widehat{Q}_r(\lambda)  ) \otimes Q_{\lambda}^{(r)}
.\end{align*}
We know that the hom is only nonzero when \(\lambda = \mu\), so this is
equal to
\begin{align*}  
\cdots
&= \widehat{L}_R(\lambda) \otimes Q_{\lambda'}^{(r)}
,\end{align*}
since in this term, the hom is just equal to \(k\). Thus we have
\begin{align*}  
\mathop{\mathrm{Soc}}_G \widehat{Q}_r(\lambda) \otimes Q_{\lambda'}^{(r)}
= \widehat{L}_r(\lambda) \otimes L(\lambda')^{(r)} = \mathop{\mathrm{Soc}}_G Q_{\lambda+ p^r \lambda'}
.\end{align*}
Since this is an injective module and the socles match, this must be an
isomorphism.

\end{proof}

\begin{corollary}[?]

Let \(\lambda \in X(T)_+\) with \(\lambda = \sum_{j=0}^s \lambda_j p^j\)
with \(\lambda_j \in X_1(T)\). Moreover, assume all \(Q_r(\mu)\) lift to
\(G\). Then there exists a \({\mathsf{G}{\hbox{-}}\mathsf{Mod}}\)
isomorphism
\begin{align*}  
Q_r(\lambda) = \bigotimes_{j=0}^s \widehat{Q}_1(\lambda_j)^{(j)}
.\end{align*}

\end{corollary}

Next time: cohomology of Frobenius kernels and relation to the nilpotent
cones, related to ABG equivalence, computing the cohomology ring for
\(G_1\).

\hypertarget{friday-november-20}{%
\section{Friday, November 20}\label{friday-november-20}}

\hypertarget{cohomology-of-frobenius-kernels}{%
\subsection{Cohomology of Frobenius
Kernels}\label{cohomology-of-frobenius-kernels}}

These predate cohomology for the small quantum group.

\begin{theorem}[Ginzburg-Kuman, 1993]

Let \(\ell > h\) be odd, then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(H^{2n}(U_\xi(g), {\mathbb{C}}) = {\mathbb{C}}[\mathcal{N}]\) for
  \(\mathcal{N}\) the nilpotent cone.
\item
  \(H^{2k+1}(U_\xi(g), G) = 0\).
\end{enumerate}

\end{theorem}

Letting \(G\) be a reductive algebra over
\(k = \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\),
recall that we have the \(r\)th Frobenius \(F^r:G\to G\), where
\(G_r \coloneqq\ker F^r\) and \(A \coloneqq\operatorname{Dist}(G_r)\).
We define
\(R = H^\cdot(G_r, k) = H^\cdot(A, K) = \operatorname{Ext} _A(k, k)\).
How do we compute this ring?

We have

\begin{center}
\begin{tikzcd}
0 \ar[r] & k \ar[r] & M_1 \ar[r] & \cdots \ar[r] & M_t \ar[r] & k \ar[r] \ar[dllll, "\cong"] & 0 \in \operatorname{Ext} _A^t(k, k) \\
0 \ar[r] & k \ar[r] & N_1 \ar[r] & \cdots \ar[r] & N_s \ar[r] & k \ar[r] & 0 \in \operatorname{Ext} _A^s(k, k) \\
\end{tikzcd}
\end{center}

We then concatenate these in \(\operatorname{Ext} _A^{s+t}(k, k)\).

The best answers occur when \(r=1\) and \(p>h\), due to results by
Friendlander-Parhsll for \(p\geq 3h-3\), and by Andersen-Jantzen for
\(p>h\).

\begin{proposition}[?]

Let \(B \subset G\) where \(B = T\rtimes U\) and
\(\mathfrak{u} = \operatorname{Lie}(U)\), \(p> h\). Then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(H^{2\cdot}(B, k) = S^0(u^*)^{(1)}\)
\item
  \(H^{2k+1}(B, k) = 0\).
\end{enumerate}

\end{proposition}

\begin{proof}[?]

The main idea: there is a spectral sequence
\begin{align*}  
E_1^{2i, j} = \qty{  
S^i(u^*)^{(1)} \otimes\Lambda^j(u^*)
}^{T_1}
\Rightarrow
\qty{ 
H^{2i+j}(U_1, k)^{T_1} = S^i(u^*) \otimes\qty{ \Lambda^j(u^*) }^{T_1} \Rightarrow H^{2i+j}(B, k)
}
\end{align*}
which is not a Grothendieck-type spectral sequence, and instead comes
from filtering a complex and taking the associated graded. Using routine
group combinatorics for \(p>h\), we can find that
\(\Lambda^\cdot(u^*)^{T_1} \cong k\). This causes the spectral sequence
to collapse.

\end{proof}

\begin{proposition}[?]

There exists a spectral sequence
\begin{align*}  
E_2^{i, j} = R^i \mathop{\mathrm{ind}}_{B/B_1}^{G/G_1} H^j(B_1, k) \Rightarrow H^{i+j}(G_1, k)
.\end{align*}

\end{proposition}

\begin{proof}[?]

\begin{fact}

We can identify
\begin{align*}  
R^j \mathop{\mathrm{ind}}_{B/B_1}^{G/G_1} S^0(u^*)^{(1)} = R^j \mathop{\mathrm{ind}}_B^G S^0(u^*) \qquad \text{for } j>0
.\end{align*}

\end{fact}

This yields

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(\mathop{\mathrm{ind}}_{B/B_1}^{G/G_1} S^0(u^*)^{(1)} = k\) (???)
\item
  \(H^{2\cdot+1}(G, k) = 0\).
\end{enumerate}

So

\begin{align*}  
H^{2\cdot}(G, k) = \qty{ \mathop{\mathrm{ind}}_B^G S^0(u^*)  }^{(1)} \coloneqq k[G \operatorname{fp}{B} u]
.\end{align*}

This yields the \textbf{Springer resolution}

\begin{center}
\begin{tikzcd}
G \operatorname{fp}{B} u \ar[r, "\pi"] & G\cdot u = \mathcal{N} \\
\pi^{-1}(O_{\text{reg}}) \ar[u, "\subset"] & O_{\text{reg}}\ar[l, "\cong"] \ar[u, "\subset"]
\end{tikzcd}
\end{center}

We can thus write
\begin{align*}  
k[G\operatorname{fp}{B} u] 
&= k[ \pi^{-1}(O_{\text{reg}}) \\
&= k[O_{\text{reg}}] \\
&= k[\mathcal{N}]
,\end{align*}
since all singularities lie in codimension strictly greater than \(2\).

\end{proof}

\begin{theorem}[F-P, A-J, 1980s]

For \(p>h\),

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(H^{2\cdot }(G, k) = k[{\mathcal{N}}]\)
\item
  \(H^{2\cdot + 1}(G, k) = 0\).
\end{enumerate}

\end{theorem}

\begin{example}[?]

Let \(G = {\operatorname{SL}}_2(k)\) and
\({\mathfrak{g}}= {\mathfrak{sl}}_2(k)\). Then for
\(x\in {\mathfrak{sl}}_2(k)\), we can write
\begin{align*}  
x = \begin{bmatrix}
a & b \\
c & -a
\end{bmatrix} && \operatorname{det}(x) = -a^2 - bc = 0
.\end{align*}
Then we have a presentation of the cohomology ring:
\begin{align*}  
H^{2\cdot}(G, k) = k[a,b,c] / \left\langle{a^2 + bc = 0}\right\rangle, && {\left\lvert {a} \right\rvert} = {\left\lvert {b} \right\rvert} = {\left\lvert {c} \right\rvert} = 2
.\end{align*}

\end{example}

\hypertarget{finite-generation-of-cohomology}{%
\subsection{Finite Generation of
Cohomology}\label{finite-generation-of-cohomology}}

A finite group scheme is essentially the same as a cocommutative Hopf
algebra.

\begin{theorem}[Friendlander-Suslin, 1997, Inv. Math]

Let \(A\) be a cocommutative Hopf algebra over \(k\). Then
\(H^{-}(A, k)\) is a finitely generated algebra

\end{theorem}

\begin{example}[?]

\(H{-}(G_r, k)\) is a finitely generated algebra.

\end{example}

\begin{problem}[Open]

Is \(H^{-}(A, k)\) a finitely generated algebra when \(A\) is a finite
dimensional Hopf algebra?

\end{problem}

\begin{problem}[Open]

Compute \(H^{-}(G_r, k)\) for \(r\geq 2\).

\end{problem}

\hypertarget{monday-november-23}{%
\section{Monday, November 23}\label{monday-november-23}}

Today: last lecture. Note that (some) videos will be available online,
see Youtube channel.

\hypertarget{cohomology-of-frobenius-kernels-1}{%
\subsection{Cohomology of Frobenius
Kernels}\label{cohomology-of-frobenius-kernels-1}}

Let \(r=1\), and consider \(H^0(G_1; k)\). For \(p > h\), we know
\(H^{2\cdot} = k[\mathcal{N}]\) and zero otherwise. What is known for
\(p<h\)?

\begin{definition}[The Restricted Nillcone]

For an arbitrary \(p\), we have the following: define
\begin{align*}  
\mathcal{N}_1({\mathfrak{g}}) = \left\{{x\in {\mathfrak{g}}{~\mathrel{\Big|}~}x^{[p]} = 0 }\right\}
,\end{align*}
where \({\mathfrak{g}}= \operatorname{Lie}(G)\).

\end{definition}

Note that \(p>h\), we have
\(\mathcal{N}_1({\mathfrak{g}}) = \mathcal{N}\). Taking
\({\mathfrak{gl}}_n\), this is matrices whose \(p\)th power is zero.
Note that the Frobenius map is still a derivation in characteristic
\(p\).

\begin{theorem}[Jantzen, 1986]

\begin{align*}  
\operatorname{mSpec}(H^\cdot(G_1; k) ) \cong \mathcal{N}_1({\mathfrak{g}})
.\end{align*}

\end{theorem}

\begin{theorem}[Carlson-Lin-Nakano-Parshall, UGA VIGRE]

For some \(G{\hbox{-}}\)orbit \(\mathcal{O}\) in \(\mathcal{N}\),
\begin{align*}  
\mathcal{N}_1({\mathfrak{g}}) = \mkern 1.5mu\overline{\mkern-1.5mu\mathcal{O}\mkern-1.5mu}\mkern 1.5mu
,\end{align*}
i.e.~it is the closure of some \(G{\hbox{-}}\)orbit. In particular,
\(\mathcal{N}_1({\mathfrak{g}})\) is irreducible.

\end{theorem}

For \(\lambda\in X(T)\), set
\begin{align*}  
\Phi_\lambda \coloneqq\left\{{ \alpha\in\Phi {~\mathrel{\Big|}~}{\left\langle {\lambda+\rho},~{\alpha {}^{ \vee }} \right\rangle} \in p{\mathbb{Z}}}\right\}
.\end{align*}

When \(p\) is good, there exists \(w\in W\) such that
\(w(\Phi_\lambda) = \Phi_J\) for some \(J \subset\Delta\).

\todo[inline]{Something about being on or off a wall, and conjugating?}

\begin{example}[?]

We can determine which \(p\) are good for each type:

\begin{itemize}
\tightlist
\item
  \(A_n\): \(p\) is always good.
\item
  \(B_n\): \(p\neq 2\)
\item
  \(C_n\): \(p\neq 2\)
\item
  \(D_n\): \(p\neq 2\)
\item
  \(E_6\): \(p\neq 2\)
\item
  \(E_7\): \(p\neq 2, 3\)
\item
  \(E_8\): \(p\neq 2, 3,5\)
\item
  \(F_4\): \(p\neq 2, 3\)
\item
  \(G_2\): \(p\neq 2, 3\)
\end{itemize}

\end{example}

\begin{conjecture}

Let \(p\) be a good and \(w(\Phi_0) = \Phi_J\) for some
\(J \subseteq \Delta\). By NPV,
\(\mathcal{N}_1({\mathfrak{g}}) = G \cdot \mu_J\). Assuming that
\(p\notdivides (X(T): {\mathbb{Z}}\Phi)\), then

\begin{itemize}
\tightlist
\item
  \(H^{2\cdot}(G; k) = k[\mathcal{N}_1({\mathfrak{g}})]\)
\item
  \(H^{2\cdot + 1}(G; k) = 0\).
\end{itemize}

\end{conjecture}

\begin{remark}

This is a natural generalization of \(p>h\). There is some situational
evidence for this to be true:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  For \(p=h-1\), we have
  \(\mathcal{N}_1({\mathfrak{g}}) = \mkern 1.5mu\overline{\mkern-1.5mu{\mathcal{O}}_{\mathrm{reg}}\mkern-1.5mu}\mkern 1.5mu\)
  and the conjecture is true.
\item
  For quantum groups, \(H^\cdot(U_q({\mathfrak{g}}); {\mathbb{C}})\) and
  it is again true.
\end{enumerate}

\todo[inline]{Something about BNPP}

\end{remark}

\begin{remark}

Some key points:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  By the Gravert-Riemenschneider theorem, for
  \(k= \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{F}}_p\mkern-1.5mu}\mkern 1.5mu\),
  \begin{align*}  
  R^{i>0} \mathop{\mathrm{ind}}_{P_J}^{C_r} S^\cdot(u_J^*) = 0
  .\end{align*}
\item
  Normality of \(\mathcal{N}_1({\mathfrak{g}})\).
\end{enumerate}

A fact about something from earlier: the ring of regular functions on
springer resolution equal to those on the nilpotent cone.

\end{remark}

\hypertarget{support-varieties}{%
\subsection{Support Varieties}\label{support-varieties}}

Very common in modern mathematics. Define \(R \coloneqq H^{-}(G_1; k)\),
which is a finitely generated algebra. Note that \(R\) acts on
\(\operatorname{Ext} _{G_1}^{-}(M, M)\), which is finitely generated
over \(R\). So what is the support of this module? Define
\(V_{G_1}(M) \coloneqq\operatorname{mSpec}\qty{R/J_M}\), where
\(J_M \coloneqq\operatorname{Ann}_R \operatorname{Ext} _{G_1}^{-}(M, M)\).

\begin{conjecture}[Jantzen, 1986]

Let \(p\) be good and \(\lambda \in X(T)_+\) be a dominant weight.
Consider \(\Phi_\lambda\) ad \(w\in W\) such that
\(w(\Phi_\lambda) = \Phi_J\) for some \(J \subseteq \Delta\). Then
\begin{align*}  
V_{G_1} (H^0(\lambda)) = G\cdot \mu_J
.\end{align*}

\end{conjecture}

\begin{remark}

Now proved! Jantzen proved for type \(A_n\), and Nakano-Parshall-Vella
proved it in general in 2002.

\end{remark}

\begin{remark}

For tilting modules, there is a conjecture for \(V_G(T(\lambda))\)
(Humphreys conjecture). Type \(A_n\) with \(p>h+1\) was verified by W.
Harelstry, and \(p=2\) by Cooper. This is still open, but is known for
quantum groups and unknown for general algebraic groups.

\end{remark}

\addsec{ToDos}
\listoftodos[List of Todos]
\cleardoublepage

% Hook into amsthm environments to list them.
\addsec{Definitions}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={definition}, numwidth=3.5em]
\cleardoublepage

\addsec{Theorems}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={theorem,proposition}, numwidth=3.5em]
\cleardoublepage

\addsec{Exercises}
\renewcommand{\listtheoremname}{}
\listoftheorems[ignoreall,show={exercise}, numwidth=3.5em]
\cleardoublepage

\addsec{Figures}
\listoffigures
\cleardoublepage


\printbibliography[title=Bibliography]


\end{document}
