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\title{
\rule{\linewidth}{1pt} \\
\textbf{
    Flag Varieties, Equivariant Cohomology, and K Theory
  }
    \\ {\normalsize Lectures by Scott Larson. University of Georgia,
Fall 2021} \\
  \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 topics in representation theory taught by
Scott Larson at the University of Georgia in Fall 2021. 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:} 2021-09-15 }
\end{flushleft}


\newpage

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

\hypertarget{first-examples-of-flagschubert-varieties-wednesday-august-18}{%
\section{First Examples of Flag/Schubert Varieties (Wednesday, August
18)}\label{first-examples-of-flagschubert-varieties-wednesday-august-18}}

\begin{remark}

Course description from Scott's syllabus:

\begin{quote}
Schubert varieties are key examples of algebraic varieties that on one
hand have an intrinsic interest and beauty, and on the other hand have
many applications to algebraic geometry, algebraic topology, and
representation theory; e.g., category \({\mathcal{O}}\), infinite
dimensional representation theory of real reductive groups, modular
representation theory, polar varieties, Chern classes, Schubert
calculus, etc.

The course goal is to understand Schubert varieties and their algebraic
geometry, equivariant cohomology, and equivariant K-theory. There are
many open problems related to basic geometry of Schubert varieties, so
we will of course not complete this goal. One of the key applications of
equivariant cohomology and equivariant K-theory of flag varieties is the
complete description of the singular locus of any Schubert variety, and
we will settle on learning this theory as our goal. This result was
originally obtained by the author of our course textbook, and is
described completely by him in Chapter XII.

The language of this result is naturally and originally described in the
ominous generality of (possibly infinite dimensional) Kac-Moody groups
-- which are becoming increasingly more important in many areas -- and
the result at the time was new even for the finite dimensional case. In
fact much recent literature on Schubert varieties is written in this
language and at the same time is new for the finite dimensional case.
\end{quote}

\end{remark}

\begin{remark}

The goal of this course: describe the singular locus of arbitrary
Schubert varieties. Note that we'll assume all varieties and schemes are
reduced!

\textbf{References:}

\begin{itemize}
\tightlist
\item
  \emph{Introduction to Lie Algebras and Representation Theory},
  Humphreys.
\item
  \emph{Representations of Semisimple Lie Algebras in the BGG Category
  \({\mathcal{O}}\)}, Humphreys.
\item
  \emph{Linear Algebraic Groups}, Humphreys.

  \begin{itemize}
  \tightlist
  \item
    \emph{Linear Algebraic Groups}, Springer.
  \end{itemize}
\item
  \emph{Kac-Moody Groups, their Flag Varieties, and their Representation
  Theory}, Shrawan Kumar.
\item
  \emph{Chries-Ginzburg.}, particularly for
  \({\mathsf{K}}{\hbox{-}}\)theory of abelian categories. See Youtube
  lectures and course notes from Geordie's course!
\item
  Brian' Conrad's notes on group schemes:
  \url{http://math.stanford.edu/~conrad/papers/luminysga3.pdf}
\item
  Björner and Brenti: \emph{Combinatorics of Coxeter Groups}
\end{itemize}

\end{remark}

\begin{remark}

First up, defining the words in the course title: flag varieties,
equivariant cohomology, \({\mathsf{K}}{\hbox{-}}\)theory.

\begin{itemize}
\tightlist
\item
  Flag variety: complete homogeneous algebraic variety, i.e.~with a
  transitive algebraic group action.
\item
  Cohomology: it suffices to work with
  \(H^*_{\mathrm{sing}}(X, A; {\mathbb{R}})\), the relative singular
  cohomology. See also Borel-Moore homology.
\item
  \({\mathsf{K}}{\hbox{-}}\)theory: The study of coherent sheaves (take
  the Grothendieck group on the category
  \(\mathsf{C} = {\mathsf{Coh}}(X)\))
\end{itemize}

\end{remark}

\begin{definition}[$T\dash$spaces]

For \(T \cong ({\mathbb{C}}^{\times})^n\) a torus, define a
\textbf{\(T{\hbox{-}}\)space} \(X\) as a space \(X\) with an action
\(T \times X\to X\) which is also an algebraic morphism.

\end{definition}

\begin{remark}

Notions of \emph{equivariance} will take into account this action. For
cohomology, we'll consider a space \(E\times^T X = (E\times X)/T\) where
\(T\) acts by \((e, x)t \coloneqq(et, t^{-1}x)\). This is not a variety,
but instead an \emph{Ind-variety}.

For \({\mathsf{K}}{\hbox{-}}\)theory, the version we'll work with is the
following:

\end{remark}

\begin{definition}[$T\dash$equivariant sheaves]

Let \(m: T\times T\to T\) be the multiplication map. For \(X\) a
\(T{\hbox{-}}\)space, a sheaf
\({\mathcal{F}}\in{\mathsf{Sh}}({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\),
\textbf{\(T{\hbox{-}}\)equivariant} iff

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  There is a given isomorphism of sheaves on \(T\times X\) written
  \(I: a^* {\mathcal{F}}\to{\operatorname{pr}}_2 {\mathcal{F}}\) where
  \({\operatorname{pr}}_2^* :T\times X\to X\) is projection onto the
  second coordinate and \(a:T\times X\to X\) is the given action map.
\item
  The pullbacks by \(\operatorname{id}\times a\) and
  \(m\times\operatorname{id}\) if the isomorphism \(I\) are given by the
  equation
  \begin{align*}
  {\operatorname{pr}}_{23}^* I \circ (\operatorname{id}_G \times a) I = (m\times\operatorname{id}_X)^* I
  .\end{align*}
\item
  There is an isomorphism \(I_{e\times X} = \operatorname{id}\) and
  \({\mathcal{F}}= a^* {\mathcal{F}}{ \left.{{}} \right|_{{e\times X}} } \xrightarrow{\sim} {\mathcal{F}}_{e\times X} = {\mathcal{F}}\).
\end{enumerate}

\end{definition}

\begin{example}[?]

Note that for \(f: X\to Y\) and \({\mathcal{F}}\in {\mathsf{Sh}}(Y)\),
then
\begin{align*}
f^{\star} {\mathcal{F}}= {\mathcal{O}}_X \otimes_{f^* {\mathcal{O}}_Y} f^* {\mathcal{F}}
.\end{align*}
For any \(T{\hbox{-}}\)space \(X\), \({\mathcal{O}}_X\) has a canonical
\(T{\hbox{-}}\)equivariant structure given by
\begin{align*}
{\operatorname{pr}}_2^* {\mathcal{O}}_X \cong {\mathcal{O}}_{T\times X} \cong a^\star {\mathcal{O}}_X
.\end{align*}

\end{example}

\begin{example}[1]

Take \(X\coloneqq{\operatorname{pt}}\cong G/G\), since any group action
is transitive and we get a complete space. This is a silly but important
example! We can take
\(H_G^* = H_G^*({\operatorname{pt}}) \coloneqq H^*_{\mathrm{sing}}({\mathsf{B}}G)\).
For \(G = {\mathbb{C}}^{\times}\), this is a polynomial ring, and for
\(T = ({\mathbb{C}}^{\times})^n\) it's just a polynomial ring in more
variables. One can then take the constant sheaf
\(\underline{{\mathbb{C}}} \in {\mathsf{Sh}}(X)\) which is
\({\mathbb{C}}\) for \(U=X\) and \(0\) otherwise.

\end{example}

\begin{example}[2]

\(X\coloneqq{\mathbb{P}}^1\) with an action by
\(G\coloneqq{\operatorname{SL}}_2({\mathbb{C}})\):

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{40pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-08-18_14-28.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

In the coordinate chart \({\left[ {z_1, z_2} \right]}\) with
\(z_2\neq 0\), we can scale \(z_2\) to 1 and set
\begin{align*}
{
\begin{bmatrix}
  {a} & {b} 
\\
  {c} & {d}
\end{bmatrix}
} {\left[ {z, 1} \right]} = {\left[ {{az+b\over cz+d}, 1} \right]} && cz +d\neq 0
.\end{align*}
Then

\begin{itemize}
\item
  \(G\curvearrowright X\) transitively, and
\item
  \(B\coloneqq{\operatorname{Stab}}_G({\left[ {0, 1} \right]})\) is a
  nontrivial Borel given by upper triangular matrices, and \(X = G/B\).
\end{itemize}

\textless-- Xournal file:
/home/zack/SparkleShare/github.com/Notes/Class\_Notes/2021/Fall/FlagVarieties/sections/figures/2021-08-18\_14-26.xoj
--\textgreater{}

\includegraphics{figures/2021-08-18_14-27-15.png}

Note that \({\mathcal{O}}_X({\mathbb{P}}^1) = {\mathbb{C}}\) by
Liouville's theorem, and \({\mathcal{O}}_X(U) \cong {\mathbb{C}}[x]\)
for \(U \subseteq {\mathbb{P}}^1\).

\end{example}

\begin{example}[3]

The Grassmannian of \(k{\hbox{-}}\)planes, given by
\begin{align*}
X^Y\coloneqq{\operatorname{Gr}}_k({\mathbb{C}}^n) = \left\{{E \subseteq {\mathbb{C}}^n {~\mathrel{\Big|}~}\dim(E) = k}\right\}
.\end{align*}
This has the structure of an algebraic group, either by taking some
transitive algebraic group action and lifting structure from the
quotient, or taking a Segre embedding. For notation, write
\({\mathbb{C}}^i \coloneqq{\operatorname{span}}_{\mathbb{C}}\left\{{e_1,\cdots, e_i}\right\}\)
for the span of the first \(i\) standard basis vectors.

\begin{itemize}
\item
  \(G\coloneqq\operatorname{GL}_n\) acts transitively by
  \(g.E \coloneqq gE\), for example by extending a basis from \(E\) to
  \({\mathbb{C}}^n\) and using that \(\operatorname{GL}_n\) sends bases
  to bases, thus sending \(E\to E'\) another \(k{\hbox{-}}\)plane.
\item
  \({\operatorname{Stab}}_G({\mathbb{C}}^2)\) are upper block-triangular
  matrices:
\end{itemize}

\textless-- Xournal file:
/home/zack/SparkleShare/github.com/Notes/Class\_Notes/2021/Fall/FlagVarieties/sections/figures/2021-08-18\_14-33.xoj
--\textgreater{}

\includegraphics{figures/2021-08-18_14-34-50.png}

Then define \(X^Y \coloneqq G/P\), noting that here \(P\) is a
parabolic.

\end{example}

\begin{remark}

Much study of Schubert varieties reduces to studies of the combinatorics
of the Weyl group. Write \(W^Y\) for the Young diagrams on an set of
\(k\times(n-k)\) blocks.

For example, for \(n=4, k=2\):

\textless-- Xournal file:
/home/zack/SparkleShare/github.com/Notes/Class\_Notes/2021/Fall/FlagVarieties/sections/figures/2021-08-18\_14-36.xoj
--\textgreater{}

\includegraphics{figures/2021-08-18_14-38-46.png}

\end{remark}

\begin{definition}[?]

For every \(\lambda\in W^Y\), define
\begin{align*}
X_\lambda^Y = \left\{{E\in X^Y {~\mathrel{\Big|}~}\forall i=1,\cdots, k,\, \dim({\mathbb{C}}^{\sum \lambda_i + i} \cap E )\geq i}\right\}
.\end{align*}

\end{definition}

\todo[inline]{Does this have a name?}

\begin{example}[?]

For \(\lambda = (1, 2)\), we have
\begin{align*}
X_\lambda^Y = \left\{{ E \in {\operatorname{Gr}}_2({\mathbb{C}}^4) {~\mathrel{\Big|}~}\dim({\mathbb{C}}^2 \cap E) \geq 1, \dim({\mathbb{C}}^4 \cap E) \geq 2 }\right\} 
.\end{align*}

\end{example}

\hypertarget{friday-august-20}{%
\section{Friday, August 20}\label{friday-august-20}}

\begin{remark}

Recall that we were discussing example 3, Grassmannians, and defined
\(W^Y\) as Young diagrams in a \(k\times (n-k)\) grid. We write
\begin{align*}
X^Y_{\lambda} = \left\{{ E \in X^Y = {\operatorname{Gr}}_k({\mathbb{C}}^n) {~\mathrel{\Big|}~}\forall 1\leq i \leq k ,\, \dim({\mathbb{C}}^{\lambda_i + i} \cap E) \geq i}\right\}
.\end{align*}

\end{remark}

\begin{example}[?]

\begin{align*}
X_{(1, 2)}^Y = \left\{{ E {~\mathrel{\Big|}~}\dim({\mathbb{C}}^2 \cap E) \geq 1, \dim({\mathbb{C}}^4 \cap E)\geq 2 }\right\} 
.\end{align*}
Note that the second condition is redundant since
\(E \subset {\mathbb{C}}^4\) is a 2-plane. Why is this a closed variety?
Perhaps the easiest way to see this is using Plucker relations. Using
more technology later, this allows follows from looking at
\(B{\hbox{-}}\)orbits and Bruhat decompositions.

\end{example}

\begin{fact}

Note that for the rank function
\({\operatorname{rank}}: \operatorname{Mat}(m\times n)\to {\mathbb{Z}}\),
one can compute the closure
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{rank}^{-1}(r)\mkern-1.5mu}\mkern 1.5mu = \operatorname{rank}^{-1}\qty{[0, r)}
.\end{align*}
Also note that
\({\operatorname{pr}}_2: {\mathbb{C}}^r\to {\mathbb{C}}^q\), we have
\(\ker ({ \left.{{{\operatorname{pr}}_2}} \right|_{{E}} }) = {\mathbb{C}}^2 \cap E\).

\end{fact}

\hypertarget{example-4-the-full-flag-variety}{%
\subsection{Example 4: The Full Flag
Variety}\label{example-4-the-full-flag-variety}}

\begin{example}[4: The Full Flag Variety (Type $A_{n-1}$)]

Define the full flag variety
\begin{align*}
X \coloneqq\left\{{ F^\bullet = \qty{0 \subseteq F^1 \subseteq F^2 \subseteq \cdots \subseteq F^{n-1} \subseteq {\mathbb{C}}^n } {~\mathrel{\Big|}~}\dim(F^k) = k}\right\} 
.\end{align*}

Write
\({\mathbb{C}}^\bullet \coloneqq\qty{0 \subseteq {\mathbb{C}}^1 \subseteq \cdots \subseteq {\mathbb{C}}^n}\)
for a distinguished basepoint.

\begin{itemize}
\tightlist
\item
  This is a complete homogeneous space,
\item
  \(GL_n\curvearrowright X\) transitively,
\item
  \({\operatorname{Stab}}_G({\mathbb{C}}^\bullet) = B\), the Borel of
  upper triangular matrices.
\item
  \(X \cong G/B\).
\end{itemize}

For \(G\) a linear algebraic group and \(B\) a closed subgroup, \(G/B\)
will generally be a variety.

\end{example}

\begin{definition}[Weyl Group]

The Weyl group is generally given by \(W = N_G(T)/T\) for \(T\) a torus.

\end{definition}

\begin{remark}

Some facts:

\begin{itemize}
\tightlist
\item
  \(N_G(T)\) is the set of permutation matrices with arbitrary nonzero
  entries.
\item
  \(W = S_n\) in general, and can be written
  \(W = \left\{{ (w(1), w(2), \cdots, w(n) ) {~\mathrel{\Big|}~}w\in S_n }\right\}\).
\item
  \(W \hookrightarrow X\) sits in the flag variety via
  \(w\mapsto c {\mathbb{C}}^\bullet\), i.e.~acting on the distinguished
  basepoint.
\end{itemize}

As an example, we can write permutation matrices in one-line notation,
using that \(w(e_i) = e_{w(i)}\):

\begin{align*}
A = {\left[ {e_4, e_1, e_2, e_3} \right]} \leadsto (4,1,2,3)
.\end{align*}

Using that \(B/B \cong {\mathbb{C}}^\bullet\) is the basepoint, we have
\(w{\mathbb{C}}^{\bullet} = wB/B \in BwB/B\).

\end{remark}

\begin{proposition}[?]

\begin{align*}
BwB / B \cong \left\{{ F^\bullet \in X {~\mathrel{\Big|}~}\forall i,j,\,\, \dim({\mathbb{C}}^i \cap F^j) \cap\dim({\mathbb{C}}^i \cap w{\mathbb{C}}^j) }\right\} 
.\end{align*}

\end{proposition}

\begin{remark}[?]

Moreover,
\(\dim({\mathbb{C}}^i \cap w{\mathbb{C}}^j) = \#\left\{{ k {~\mathrel{\Big|}~}k\leq i, w(k) \leq j}\right\}\).
Just compute
\(\left\langle{e_1}\right\rangle \cap w\left\langle{e_1}\right\rangle = \left\langle{e_4}\right\rangle = 0\)
for entry \(1, 1\), and continue:
\begin{align*}
\begin{bmatrix}
0 & 1 & 1 & 1 \\
0 & 1 & 2 & 2 \\
0 & 1 & 2 & 3 \\
1 & 2 & 3 & 4 \\
\end{bmatrix}
.\end{align*}

Now check that counting
\(\left\{{k {~\mathrel{\Big|}~}k\leq j,\, w(k) \leq i}\right\}\) yields
the same entries in the \(i, j\) spot, and thus the same matrix.

\end{remark}

\hypertarget{combining-examples-3-and-4}{%
\subsection{Combining Examples 3 and
4}\label{combining-examples-3-and-4}}

\begin{remark}

There is a map
\begin{align*}
\pi: X &\to X^Y \\
F^\bullet &\mapsto F^k
,\end{align*}
which is equivalently sending a Borel to its corresponding parabolic,
and geometrically corresponding to sending \(T{\hbox{-}}\)fixed points
to \(T{\hbox{-}}\)fixed points. This induces a map \(W\to W^Y\), and
since \(W\cong S_n\), this is sending a Young diagram to a partition.

\begin{itemize}
\tightlist
\item
  This is \(G{\hbox{-}}\)equivariant for
  \(G\coloneqq\operatorname{GL}_n\)
\item
  \(\pi(w) = \lambda\), so there is a map
  \(w\to \left\{{w(1), \cdots, w(k)}\right\} = \left\{{ \lambda_1 - 1, \lambda_2 - 2, \cdots, \lambda_k - k}\right\}\).
\end{itemize}

\end{remark}

\begin{example}[?]

Given \(\lambda\) and \(1\leq i \leq k\), let \(w(i) = \lambda_i + i\)
and extend \(w\) by filling in the remaining numbers in increasing
order, so \(w(k+1) < w(k+2) < \cdots < w(n)\). For example, take
\((1, 2)\mapsto w = (2, 4 {~\mathrel{\Big|}~}1, 3)\), recalling that
\((1, 2)\) has this form:

\includegraphics{figures/2021-08-20_14-32-17.png}

One could also do \(w_{\max} = (4, 2 {~\mathrel{\Big|}~}3, 1)\).

\end{example}

\begin{remark}

Note that the Hasse diagrams under a given diagram give the closure
relations under \(B{\hbox{-}}\)orbits: For \(\lambda = (1, 2)\), the
\(B{\hbox{-}}\)orbits in \(X_\lambda^Y\) are given by the following:

\includegraphics{figures/2021-08-20_14-37-13.png}

We get \(BwP/P = {\mathbb{C}}^{\ell(w^r)}\), and we in fact get a CW
structure. Since
\(H_{\mathrm{sing}}^2(X_\lambda^Y; {\mathbb{Z}}) \neq H_{\mathrm{sing}}^4(X_\lambda^Y; {\mathbb{Z}})\),
this doesn't satisfying Poincare duality, so it can not be a smooth
manifold. So what is the singular locus?

\end{remark}

\begin{remark}

The open element is not in the singular locus. Note that \(P\) acts on
\(X_\lambda^Y\), where \(P\) is defined as

\includegraphics{figures/2021-08-20_14-40-59.png}

One can determine that the singular locus is the single point
\(\left\{{{\mathbb{C}}^2}\right\}\) corresponding to the empty diagram:

\includegraphics{figures/2021-08-20_14-42-09.png}

\end{remark}

\hypertarget{lecture-2-monday-august-23}{%
\section{Lecture 2 (Monday, August
23)}\label{lecture-2-monday-august-23}}

\hypertarget{a-lightning-introduction-to-groups-and-representations}{%
\subsection{A Lightning Introduction to Groups and
Representations}\label{a-lightning-introduction-to-groups-and-representations}}

\begin{remark}

Throughout, \emph{finite type} means finitely generated over the base
field.

\end{remark}

\begin{remark}

Which \(G\) are important for equivariant cohomology of the flag
variety, and equivariant \({\mathsf{K}}{\hbox{-}}\)theory. We'll
consider only connected reductive groups, and work over
\(k \coloneqq{\mathbb{C}}\).

\end{remark}

\begin{definition}[Pertaining to Linear Algebraic Groups]

\envlist

\begin{itemize}
\item
  A group \(G \in {\mathsf{Alg}}{\mathsf{Grp}}\) be is a \textbf{linear
  algebraic group} if

  \begin{itemize}
  \tightlist
  \item
    The coordinate ring \({\mathbb{C}}[G]\) is a reduced (so no nonzero
    nilpotents) \({\mathbb{C}}{\hbox{-}}\)algebra of finite type.
  \item
    \(G\) is a group where multiplication \(m:G^{\times 2}\to G\) and
    inverseion \(i:G\to G\) are algebraic morphisms
  \end{itemize}
\item
  A \textbf{maximal torus} of \(G\) is a torus not properly contained in
  any other torus of the form \(({\mathbb{C}}^{\times})^{\times n}\).
\item
  A \textbf{Cartan} subgroup is the centralizer of a maximal torus. Note
  that maximal torii are the same as Cartans in the connected reductive
  case.
\item
  \(G\) is \textbf{unipotent} if every representation has a nonzero
  fixed vector.
\item
  The \textbf{unipotent radical} \(R_u(G)\leq G\) is a maximal closed
  connected normal subgroup of \(G\).
\item
  \(G\) is \textbf{reductive} iff \(R_u(G) = \left\{{e}\right\}\).
\end{itemize}

\end{definition}

\begin{proposition}[?]

To study \({\mathsf{Rep}}(G)^{{\mathrm{irr}}}\) for
\(G\in {\mathsf{Alg}}{\mathsf{Grp}}\) linear, we can assume that \(G\)
is reductive.

\end{proposition}

\begin{proof}[?]

Let \(V\in {\mathsf{Rep}}(G)^{{\mathrm{irr}}}\), we'll show that the
unipotent radical acts trivially. Then \(V\) is the data of

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(G\to \operatorname{GL}(V)\) for some \(V\), a morphism of varieties
  and algebraic groups
\item
  There is an action map \(G\times V\to V\).
\end{enumerate}

Let \(V_0 = \mathrm{Fix} (R_u(G)) \subseteq V\) be the fixed points of
\(R_u(G)\), by restricting the \(G\) action to an \(R_u(G)\leq G\)
action by a subgroup. We know \(V_0 \neq 0\), and we have for every
\(g\in G, r\in R_u(G), v\in V_0\). We'd like to show \(V_0 = V\), which
means that \(R_u(G)\) acts trivially. So we'll show \(r\) fixes every
\(gv\):
\begin{align*}
r(gv) = g(g^{-1}r g)v \in g R_u(G) v = gv
,\end{align*}
using that \(R_u(G)\) fixes \(v\). So \(V_0\) is \(G{\hbox{-}}\)stable,
and since \(V_0\) is irreducible and \(V\) is irreducible, we get
equality.

\end{proof}

\begin{remark}

So \(R_u(G)\) won't matter for irreducible representations, or in turn
for equivariant \({\mathsf{K}}{\hbox{-}}\)theory, and we can assume
\(R_u(G) = \left\{{e}\right\}\) is trivial. If \(G\) is not reductive,
just replace it with \(R/R_u(G)\), which is a reductive linear algebraic
group when \(G\) is a linear algebraic group since
\(R_u(G) {~\trianglelefteq~}G\).

Next question: how can we relate compact groups to complex reductive
groups?

\end{remark}

\begin{remark}

Let \(K \in \operatorname{Lie}{\mathsf{Grp}}\) be compact, and set
\({\mathbb{C}}[K]\) to be the \({\mathbb{C}}{\hbox{-}}\)span of matrix
coefficients of finite dimensional representations of \(K\). For \(V\) a
finite-dimensional representation of \(K\) (just a continuous
representation of a compact group), define
\begin{align*}
\phi: V {}^{ \vee }\otimes_{\mathbb{C}}V &\to {\mathbb{C}}[K] \\
f\otimes v &\mapsto \qty{k \xrightarrow{\phi_{f, v}}  f(kv)}
.\end{align*}

\end{remark}

\begin{fact}

\({\mathbb{C}}[K]\) is a finite type reduced algebra. Such algebras
correspond to an affine variety, i.e.~it is the ring of functions on
some affine variety. Thus \({\mathbb{C}}[K] = {\mathbb{C}}[G]\) for
\(G\in {\mathsf{Aff}}{\mathsf{Var}}_{/ {{\mathbb{C}}}}\) where
\(K \subseteq G\).

\end{fact}

\begin{theorem}[Chevalley]

\envlist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(G\) is a \emph{reductive} algebraic group.
\item
  Every locally finite continuous representation of \(K\) extends
  uniquely to an algebraic representation of \(G\), and every algebraic
  representation of \(G\) restricts to a locally finite representation
  of \(K\).
\end{enumerate}

\end{theorem}

\begin{remark}

So despite \({\mathbb{C}}[G]\) being infinite dimensional, every
representation is contained in some finite dimensional piece. Note that
there is an equivalence of categories between algebraic and compact
groups, but there are differences: e.g.~there are no irreducible
infinite dimensional representations of compact groups.

\begin{quote}
Side note, see stuff by David Vogan!
\end{quote}

\end{remark}

\begin{remark}

The next result reduces representations to Cartans, which are
\emph{almost} tori, and is along the lines of what Langlands was
originally thinking about.

\end{remark}

\begin{theorem}[Cartan-Weyl]

There is a bijection
\begin{align*}
\widehat{G} \coloneqq\left\{{\substack{
  \text{Irreducible representations} \\
  \text{of }G
}}\right\}
&\rightleftharpoons
\left\{{\substack{
  \text{Irreducible dominant representations} \\
  \text{of a Cartan subgroup } H\leq G
}}\right\}
\end{align*}

Moreover,

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  If \(G\) is finite,
  \(\left\{{e}\right\} = B \supseteq = \left\{{e}\right\}\), so there is
  no reduction in this case, noting that the centralizer ends up being
  the whole group.
\item
  If \(G\) is connected reductive, then \(T=H\) and there reduce to
  dominant characters of a torus.
\end{enumerate}

\end{theorem}

\begin{remark}

See David Vogan's orange book on unitary representations of real
reductive groups.

\end{remark}

\begin{exercise}[?]

Try proving this directly!

\end{exercise}

\begin{definition}[Dominant characters]

Define
\begin{align*}
X(T) \coloneqq\left\{{T \xrightarrow{f} {\mathbb{C}}^{\times}{~\mathrel{\Big|}~}f \text{ is algebraic}}\right\}
,\end{align*}
which is a moduli of irreducible representations of \(G\). Then
\begin{align*}
X(T) \supseteq D_{\mathbb{Z}}\coloneqq\left\{{\chi \in X {~\mathrel{\Big|}~}\chi \text{ is dominant for } B}\right\}
.\end{align*}
Note that this may make more sense after seeing root systems.

\end{definition}

\begin{remark}

Given \(\lambda \in D_{\mathbb{Z}}\), define a
\(G{\hbox{-}}\)equivariant line bundle on the flag variety as
\({\mathcal{L}}(\lambda) \coloneqq(G\times{\mathbb{C}}_{-\lambda})/B\),
where \((-\lambda)t \coloneqq\lambda(t)^{-1}\). This can be extended to
a representation of \(B\) by
\begin{align*}
B \to B/R_u(B) \cong T \xrightarrow{\lambda} {\mathbb{C}}^{\times}
.\end{align*}
This makes sense thinking of a Borel as upper-triangular matrices, tori
as diagonal matrices, and unipotent as strictly upper triangular. So we
can extend representations by making them trivial on a normal subgroup?

\todo[inline]{Check}

We refer to \(\lambda\) as the map and \({\mathbb{C}}_{\lambda}\) as the
vector space in the representation \(G\to \operatorname{GL}(V)\). Note
that \(B\) acts on the right of \(G\times {\mathbb{C}}_{-\lambda}\) by
\begin{align*}
(g, z)b \coloneqq(gb, b^{-1}z) \coloneqq(gb, \lambda(b)^{-1}z)
.\end{align*}

\end{remark}

\begin{fact}

\({\mathcal{L}}(\lambda)\) is an algebraic variety.

\end{fact}

\hypertarget{more-broad-overview-wednesday-august-25}{%
\section{More Broad Overview (Wednesday, August
25)}\label{more-broad-overview-wednesday-august-25}}

\begin{remark}

We'll assume background in affine varieties, but not necessarily
sheaves. Today's material: see Springer.

\end{remark}

\begin{definition}[Ringed Spaces]

Let \(X\in {\mathsf{Top}}\), then a \textbf{ringed space} is the data of
\(X\) and for all \(U\in {\mathsf{Open}}(X)\) an assignment
\({\mathcal{O}}(U) \in {\mathsf{Alg}}_{{\mathbb{C}}}\) a
\({\mathbb{C}}{\hbox{-}}\)algebra of complex functions satisfying
\emph{restriction} and \emph{extension}, also known as a sheaf of
\({\mathbb{C}}{\hbox{-}}\)valued functions. A \textbf{morphism} of
ringed spaces \(\xi:X\to Y\) is a continuous function such that for all
\(W\in {\mathsf{Open}}(Y)\), one can form the pullback
\begin{align*}
\xi_W^*f: \xi^{-1}(W) \xrightarrow{\xi} W \xrightarrow{f} {\mathbb{C}}
,\end{align*}
and we require that there is a well-defined induced map
\(\xi_W^*: {\mathcal{O}}_Y(W) \to {\mathcal{O}}_X(\xi^{-1}(W))\).

\end{definition}

\begin{example}[?]

For \(X\) an affine variety, the sheaf \({\mathcal{O}}_X\) of regular
functions satisfies this property. Note that \({\mathcal{O}}\) can be an
arbitrary sheaf though, not necessarily just regular functions.

\end{example}

\begin{definition}[Prevariety]

A \textbf{prevariety} \(X\) is a quasicompact space \(X\) such that
every \(x\in X\) admits a neighborhood \(U \subseteq X\) such that
\((U, \mathop{\mathrm{Res}}({\mathcal{O}}_X, U))\) is isomorphic to an
affine variety. A prevariety is a \textbf{variety} if it is additionally
separated, so \(\Delta_X \subseteq X^{\times 2}\) is closed.

\end{definition}

\begin{remark}

Last time we said that \({\mathcal{L}}(\lambda)\) is an \emph{algebraic
variety}, so it satisfies the above definitions.

\end{remark}

\begin{remark}

From now on \(G\) will be a connected reductive group.
\(\pi: G\to {\mathcal{L}}(\lambda)\) will always be the map from the
group to the flag variety.

\end{remark}

\begin{remark}

Let \(X \in {\mathsf{Alg}}{\mathsf{Var}}_{/ {{\mathbb{C}}}}\) and
\(H\in {\mathsf{Alg}}{\mathsf{Grp}}\) be linear where
\(H\curvearrowright X\). Then \(X/H\) is a quotient in
\({\mathsf{Top}}\), by just taking the quotient topology. Let
\(\rho: X\to X/H\) be the projection, then define the ring of functions
as
\begin{align*}
{\mathcal{O}}_{X/H}(U) \coloneqq\left\{{f\in \operatorname{Hom}(U, {\mathbb{C}}) {~\mathrel{\Big|}~}\mathop{\mathrm{Res}}(f \circ \rho, \rho^{-1}(U) ) \in {\mathcal{O}}_X(\rho^{-1}(U))}\right\}
.\end{align*}
In this way \({\mathcal{O}}_{X/H}(U)\) can be identified with
\(H{\hbox{-}}\)invariant functions \({\mathcal{O}}_X(\rho^{-1}(U))^H\).
This makes \(X/H\) a ringed space, which is often (but not necessarily)
an algebraic variety.

\end{remark}

\begin{example}[?]

This is not always an algebraic variety, e.g.~taking
\({\mathbb{C}}^{\times}\curvearrowright{\mathbb{C}}\) by multiplication.
This yields two orbits (0 and everything else) and isn't a variety.

\end{example}

\begin{remark}

If \(\pi: G\to G/H\) has local sections, then
\((G\times X)/H \in {\mathsf{Alg}}{\mathsf{Var}}\) using
\((g, x)h \coloneqq(gh, h^{-1}x)\). Note that this is a fiber bundle for
the Zariski topology, and doesn't have local sections (contrasting the
analytic topology).

\end{remark}

\begin{claim}

The map \(\pi: G\to G/B\) has local sections (but no global sections).

\end{claim}

\begin{remark}

Side note: we have the Bruhat decomposition
\(G = {\textstyle\coprod}_{w\in W} BwB\) as a partition into double
cosets, quotienting by an action of \(B\times B\). The theorem is that
these are parameterized by the Weyl group.

\end{remark}

\begin{remark}

Let \(B = TU\) where \(T\) is a torus and \(U\) unipotent (so upper
triangular, ones along the diagonal) and set \(U^-\) to be the
\emph{opposite unipotent radical} (e.g.~lower triangular, ones along
diagonal). Define a map
\begin{align*}
\phi: U^i \times B &\to G \\
(\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu, b) &\mapsto \mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mub^{-1}
.\end{align*}
Then \(\operatorname{im}(\phi) = U^- B\), and \(\phi\) is injective
since \(U^- \cap B = \left\{{e}\right\}\). The argument on matrices
holds more generally: \(B\) are the upper triangular matrices and
\(U^-\) has ones on the diagonal, so these intersect only at the
identity. \(\phi\) is an open embedding: one can show that the
derivative is surjective:
\begin{align*}
d\phi(1, 1): {\mathfrak{u}}^- \times {\mathfrak{b}}&\to {\mathfrak{y}}\\
(x, y) &\mapsto x-y
.\end{align*}
Rewriting the target as
\({\mathfrak{u}}^- \oplus {\mathfrak{h}}\oplus {\mathfrak{u}}^+\) where
\({\mathfrak{b}}= {\mathfrak{h}}\oplus {\mathfrak{u}}^+\), one can find
preimages of any element.

Define a local section: \(\sigma: U\to G\) where \(U \subseteq G/B\).
Use the composite \(U^- \times B\to G \to G/B \supseteq U^-\) to view
\(U^-\) as a subset of the flag variety. An explicit formula for section
is the following:
\begin{align*}
\sigma(\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu) \coloneqq(\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu, 1) \in U^- \times B \subseteq G
.\end{align*}
Although this only constructs a section for one open set, translating by
elements of \(g\) yields an open cover, and everything is equivariant.

\end{remark}

\begin{remark}

Using this, \((G\times X)/B\) is always an algebraic variety, since
\(G\to G/B\) always has local sections. For other groups, \(X\)
quasiprojective will also make the quotient algebraic, but the proof is
more difficult. However it still involves constructing local sections.
It turns out that
\(G{ \underset{\scriptscriptstyle {B} }{\times} }X\to G/B\) is a locally
trivial fiber bundle.

\end{remark}

\begin{remark}

A note on notation: \((G\times X)B\) is sometimes written
\(G{ \underset{\scriptscriptstyle {B} }{\times} } X\) (as above), but
this is \emph{not} a fiber product. In this notation,
\({\mathcal{L}}(\lambda) = G{ \underset{\scriptscriptstyle {B} }{\times} } {\mathbb{C}}_{- \lambda}\).
Note that this is a line bundle on \(G/B\), so we can take sections.

\end{remark}

\begin{theorem}[Borel-Weil]

\envlist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  There is a correspondence
  \begin{align*}
    H^0(G/B; {\mathcal{L}}(\lambda)) \cong \left\{{f:G\to {\mathbb{C}}{~\mathrel{\Big|}~}f(g) = bf(gb) }\right\} && G \in {\mathbb{C}}[G]
    .\end{align*}
\end{enumerate}

A section
\(\sigma: G/B \to G{ \underset{\scriptscriptstyle {B} }{\times} } {\mathbb{C}}_{- \lambda}\)
gets sent to \(\sigma(gB/B) = [g, f(g)]\). Use that the quotient acts
like a tensor over \(B\), so
\begin{align*}
  gB/B = gbB/B = [gb, f(gb)] = [g, b f(gb)]
  .\end{align*}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  \(H^0(G/B; {\mathcal{L}}(\lambda)) = L_{\lambda} {}^{ \vee }\) for
  \(\lambda\) a dominant character in \(D_{\mathbb{Z}}\), where
  \(L_\lambda\) is the irreducible finite dimensional representation of
  \(G\) with highest weight \(\lambda\). Note that in the finite case,
  we have \(L_\lambda {}^{ \vee }= L_{w_0 \lambda}\), but in the
  Kac-Moody case one doesn't have \(w_0\).
\end{enumerate}

\end{theorem}

\begin{example}[?]

For \(\lambda = 0 \in X(T)\) a character, we get
\begin{align*}
\left\{{f:G\to {\mathbb{C}}{~\mathrel{\Big|}~}f(g) = f(gb)}\right\} = {\mathbb{C}}[G/B] = {\mathcal{O}}_{G/B}(G/B)={\mathbb{C}}
.\end{align*}

\end{example}

\begin{remark}

Chapter 1 of Kumar, Cartan matrices.

\end{remark}

\hypertarget{starting-kumar-friday-august-27}{%
\section{Starting Kumar (Friday, August
27)}\label{starting-kumar-friday-august-27}}

\hypertarget{definition-of-kac-moody-algebras}{%
\subsection{1.1: Definition of Kac-Moody
Algebras}\label{definition-of-kac-moody-algebras}}

\begin{definition}[Realization]

Let \(A\in \operatorname{Mat}(\ell\times\ell, {\mathbb{C}})\) be rank
\(r\). A \textbf{realization} of \(A\) is a triple
\(({\mathfrak{h}}, \pi, \pi {}^{ \vee })\) where
\(h\in{\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\),
\(\pi = \left\{{\alpha_1, \cdots, \alpha_\ell}\right\} \subseteq {\mathfrak{h}} {}^{ \vee }\)
are column vectors, and
\(\left\{{ \alpha_1 {}^{ \vee }, \cdots, \alpha_\ell {}^{ \vee }}\right\} \subseteq {\mathfrak{h}}\)
are row vectors are indexed sets satisfying

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\pi, \pi {}^{ \vee }\) are linearly independent sets.
\item
  \(\alpha_j( \alpha_i {}^{ \vee }) = a_{i, j}\)
\item
  \(\ell - r = \dim_{\mathbb{C}}({\mathfrak{h}}) - \ell\)
\end{enumerate}

\end{definition}

\begin{proposition}[?]

There exists a realization of \(A\) that is unique up to isomorphism.
Moreover, realizations of \(A, B\) are isomorphic iff \(B\) is similar
to \(A\) via a permutation of the index set.

\end{proposition}

\begin{proof}[?]

Assume \(A\) is of the form
\begin{align*}
A = 
\begin{bmatrix}
A_1  
\\
A_2
\end{bmatrix}
,\end{align*}
where \(A_1\) is \(r\times \ell\) block where
\(\operatorname{rank}A_1 = r\) and \(A_2\) is \(l-r\times \ell\) Set
\begin{align*}
C \coloneqq
\begin{bmatrix}
A_1 & 0 
\\
A_2 & I_{\ell-r}
\end{bmatrix}\in \operatorname{Mat}(\ell \times (2\ell - r))
.\end{align*}
For \({\mathfrak{h}}= {\mathbb{C}}^{2\ell - r}\), set
\(\alpha_1, \cdots, \alpha_\ell\) to be the first \(\ell\) coordinate
functions \(\alpha_1 {}^{ \vee },\cdots\) as the rows of \(C\). This is
a realization.

Conversely, given a realization
\(({\mathfrak{h}}, \pi, \pi {}^{ \vee })\), we can produce a matrix:
complete \(\pi\) to a basis of \({\mathfrak{h}} {}^{ \vee }\). This can
done in such a way that
\(\alpha_j(\alpha_i {}^{ \vee }) = [A_1, B; A_2, D^{-1}]\in \operatorname{Mat}(\ell \times 2\ell -r\).
Using column operations, i.e.~multiplication on the right, this can be
mapped to \([A_1, 0; A_2, I]\).

\end{proof}

\begin{definition}[Free Lie algebra generated by a vector space]

Let \(V\in {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) and
\(T^\bullet(V)\) be its (associative) tensor algebra. Set
\([ab] = ab-ba\) and take \(F(V) \subseteq T(V)\) to be the free Lie
algebra generated by \(T^1(V)\). We call \(F(V)\) the \textbf{free Lie
algebra generated by \(V\)}. There is a universal property: for any
linear hom \(\theta: V\to {\mathfrak{sl}}\), there is a commuting
diagram

\begin{center}
\begin{tikzcd}
    V && {\mathfrak{s}} \\
    \\
    & {F(V)}
    \arrow["\theta", from=1-1, to=1-3]
    \arrow["F"', from=1-1, to=3-2]
    \arrow["{\exists\tilde \theta}"', dashed, from=3-2, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJWIl0sWzIsMCwiXFxtYXRoZnJha3tzfSJdLFsxLDIsIkYoVikiXSxbMCwxLCJcXHRoZXRhIl0sWzAsMiwiRiIsMl0sWzIsMSwiXFxleGlzdHNcXHRpbGRlIFxcdGhldGEiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=}{Link
to Diagram}
\end{quote}

Note that \(U(F(V)) = T(V)\).

This can be constructed as
\begin{align*}
{\mathfrak{h}}\oplus \left\langle{e_1, \cdots, e_\ell}\right\rangle \bigoplus \left\langle{f_1, \cdots, f_\ell}\right\rangle / \sim
\\ \\
\sim \coloneqq
\begin{cases}
[e_i f_i] = \delta_{ij} \alpha_i {}^{ \vee }& i, j = 1,\cdots, \ell 
\\
[hh'] = 0 & h,h'\in{\mathfrak{h}}
\\
[he_i = \alpha_i(h)e_i & 
\\
[hf_i = \alpha_i(h)f_i & i=1,\cdots,\ell, h\in {\mathfrak{h}}
\end{cases}
.\end{align*}
Then set \(\tilde {\mathfrak{g}}(A) \coloneqq F(V)/\sim\) We'll find
that this only depends on the realization of \(A\).

\end{definition}

\begin{definition}[Generalized Cartan Matrices]

A matrix \(A = (\alpha_{ij})\) is a \textbf{generalized Cartan matrix
(GCM)}:

\begin{itemize}
\tightlist
\item
  \(\alpha_{ii} = 2\)
\item
  \(\alpha_{ij} \leq 0, i\neq j\)
\item
  \(\alpha_{ij} = 0\) if \(\alpha_{ji} =0\)
\end{itemize}

\end{definition}

\begin{definition}[Kac-Moody Lie Algebras]

The \textbf{Kac-Moody Lie algebra} is defined by
\({\mathfrak{g}}\coloneqq{\mathfrak{g}}(A) \coloneqq\tilde{\mathfrak{g}}(A)/\sim\),
where we mod out by the \textbf{Serre relations}:
\begin{align*}
(\operatorname{ad}e_i)^{1 - a_{ij}}(e_j) &= 0\\
(\operatorname{ad}f_i)^{1 - a_{ij}}(f_j) &= 0
.\end{align*}

\end{definition}

\begin{remark}

\envlist

\begin{itemize}
\tightlist
\item
  There is an injection \({\mathfrak{h}}\hookrightarrow{\mathfrak{g}}\),
  so we refer to \({\mathfrak{h}}\) as the \textbf{Cartan subalgebra}.
\item
  The \(e_i, f_i\) are \textbf{Chevalley generators}.
\item
  The \textbf{nilradicals} are
  \({\mathfrak{n}}\coloneqq\left\langle{\left\{{e_1,\cdots,e_\ell}\right\}}\right\rangle\)
  and
  \({\mathfrak{n}}^- \coloneqq\left\langle{ \left\{{f_1, \cdots, f_\ell}\right\}}\right\rangle\).
\item
  \({\mathfrak{b}}\coloneqq{\mathfrak{h}}\oplus {\mathfrak{n}}\) is the
  \textbf{standard Borel}.
\item
  \({\mathfrak{b}}^- \coloneqq{\mathfrak{h}}\oplus {\mathfrak{n}}\)
\item
  \(\tilde {\mathfrak{n}}, \tilde {\mathfrak{n}}^-, \tilde {\mathfrak{b}}, \tilde {\mathfrak{b}}^-\)
  can similarly be defined for \(\tilde {\mathfrak{g}}\).
\end{itemize}

\end{remark}

\begin{remark}

A big theorem from algebraic groups: a connected reductive group \(G\)
corresponds to a root
datum\(({\mathfrak{g}}, \left\{{\alpha_i}\right\}_{i\leq \ell}, \left\{{ \alpha_i {}^{ \vee }}\right\}_{i\leq \ell} )\)
where \(\alpha_i, \alpha_i {}^{ \vee }\in {\mathbb{Z}}^n\) such that
\(a_{ij} \coloneqq{\left\langle {\alpha_i},~{\alpha_i {}^{ \vee }} \right\rangle}\)
form a Cartan matrix \(A \coloneqq(a_{ij})\).

\end{remark}

\begin{example}[?]

Consider pairs of \(K, G\) where \(G\) is the complexification of \(K\):

\begin{itemize}
\tightlist
\item
  \({\mathsf{Sp}}_n \leadsto {\mathsf{Sp}}_{2n}({\mathbb{C}})\),
  \(Z(G) = {\mathbb{Z}}/2\) for \(n\geq 1\)
\item
  \({\operatorname{SU}}_n \leadsto {\operatorname{SL}}_n({\mathbb{C}})\),
  \(Z(G) = {\mathbb{Z}}/4n\) for \(n\geq 3\)
\item
  \({\operatorname{Spin}}_n \leadsto {\operatorname{Spin}}_n({\mathbb{C}})\),
  \(Z(G) = ({\mathbb{Z}}/2)^2\), \(n\geq 8\) even
\item
  \(F_4\), \(Z(G) = {\mathbb{Z}}/4\) for \(n\geq 7\) odd
\item
  \(G_2\)
\item
  \(E_6\)
\item
  \(E_7\)
\item
  \(E_8\)
\end{itemize}

Here we take the simply connected groups for the last 5, and the last 4
have cyclic centers.

\end{example}

\begin{theorem}[?]

There exist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Simple, simply connected, connected groups \(G_1, \cdots, G_k\),
\item
  A finite central subgroup \(F \subseteq \prod G_i \times T'\) where
  \(T'\) is a (not necessarily maximal) torus,
\end{enumerate}

such that \(G\cong (\prod G_i \times T')/F\). All connected reductive
groups arise this way!

\end{theorem}

\begin{example}[?]

Let
\(G\coloneqq\operatorname{GL}_n = {\operatorname{SL}}_n \cdot {\mathbb{C}}^{\times}\),
and they intersect at roots of unity, so
\begin{align*}
\operatorname{GL}_n = ({\operatorname{SL}}_n \times {\mathbb{C}}^{\times}) / \left\langle{\zeta_n I_n, \zeta_n^{-1}}\right\rangle
.\end{align*}
The map (in the reverse direction) is \((g, z)\mapsto gz\), and if
\(gz= I\) in \(\operatorname{GL}_n\) then \(g = \zeta_n^k I_n\) and
\(z = \zeta_k^{-1}\).

\end{example}

\begin{remark}

Assume \(G\) is semisimple, simply connected, and connected. Then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  The equivariant cohomology is
  \begin{align*}
  H^*_T(G/B; {\mathbb{Q}}) \cong S_{\mathbb{Q}}\otimes_{S_{\mathbb{Q}}^W} S_{\mathbb{Q}}
  \end{align*}
\item
  The equivariant \({\mathsf{K}}{\hbox{-}}\)theory
  \begin{align*}
  K^T(G/B) = A(T) \otimes_{A(T)^W}A(T)
  \end{align*}
\end{enumerate}

Note that
\begin{align*}
W &= N_G(T)/T \\
S &= S({\mathfrak{h}} {}^{ \vee }), && \pi \subseteq {\mathfrak{h}} {}^{ \vee }\\
A(T) &= {\mathbb{Z}}[X(T)]
.\end{align*}

\end{remark}

\begin{remark}

Think about semisimple, simply connected, and connected groups most of
the semester.

\end{remark}

\hypertarget{kac-moody-groups-monday-august-30}{%
\section{Kac-Moody Groups (Monday, August
30)}\label{kac-moody-groups-monday-august-30}}

\begin{remark}

See exercises in first two sections, 1.1 and 1.2. See also the proof of
the Borel-Weil theorem.

\end{remark}

\hypertarget{root-space-decompositions}{%
\subsection{1.2: Root Space
Decompositions}\label{root-space-decompositions}}

\begin{remark}

Starting with a generalized Cartan matrix \(A\), we produced a Lie
algebra \(\tilde{\mathfrak{g}}(A)\) by taking the free Lie algebra and
modding out by certain relations. This algebra only depended on the
realization of \(A\), namely \(({\mathfrak{g}}, \pi, \pi {}^{ \vee })\),
which we thought of as
\(({\mathfrak{g}},{\mathfrak{h}} {}^{ \vee }, {\mathfrak{h}})\),
yielding \({\mathfrak{g}}(A)\) modulo Serre relations.

\end{remark}

\begin{definition}[Root Lattice]

Define
\begin{align*}
Q &\coloneqq{\mathbb{Z}}\pi \subseteq {\mathfrak{h}} {}^{ \vee }&& \text{the root lattice }\\
Q^+ &\coloneqq{\mathbb{Z}}_{\geq 0} \pi \subseteq {\mathfrak{h}} {}^{ \vee }\\
{\mathfrak{g}}_\alpha &\coloneqq\left\{{x\in g {~\mathrel{\Big|}~}\forall h\in{\mathfrak{h}}, [hx] = \alpha(h) x}\right\} && \text{for }\alpha\in{\mathfrak{h}} {}^{ \vee }
.\end{align*}

\end{definition}

\begin{theorem}[?]

\envlist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \({\mathfrak{g}}= {\mathfrak{n}}^- \oplus {\mathfrak{h}}\oplus {\mathfrak{n}}\),
  which are all nonzero.
\item
  \({\mathfrak{n}}^{\pm \alpha} = \bigoplus_{\alpha\in Q^+\setminus\left\{{0}\right\}} {\mathfrak{g}}_{\pm \alpha}\).
\item
  \(\dim_{\mathbb{C}}{\mathfrak{g}}_\alpha < \infty\).
\item
  \({\mathfrak{n}}\coloneqq\left\langle{e_1,\cdots, e_\ell}\right\rangle\)
  subject \emph{only} to the Serre relations, i.e.~no additional
  relations are needed for this subalgebra.
\end{enumerate}

\end{theorem}

\begin{proof}[?]

First step: prove for \(\tilde {\mathfrak{g}}\) and put a tilde on
everything appearing in the theorem statement.

Let \(\left\{{v_1, \cdots, v_\ell}\right\}\) be a basis for \(V\) and
fix \(\lambda \in {\mathfrak{h}} {}^{ \vee }\). Define an action of
generators of \(\tilde {\mathfrak{g}}\) on \(T(V)\) in the following
way:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\alpha:\) Set \(f_i(\alpha) \coloneqq v_i\otimes a\) for
  \(a\in T(V)\)
\item
  \(\beta:\) set
  \(h(1) \coloneqq{\left\langle {\lambda},~{h} \right\rangle}1 \coloneqq\lambda(h)\cdot 1\),
  and inductively on \(s\) set
  \begin{align*}
  h(v_j \otimes a) \coloneqq- {\left\langle {\alpha_j},~{h} \right\rangle} v_j \otimes a + v_j \otimes h(a) && a\in T^{s-1}(V), h\in {\mathfrak{h}}, 1\leq j \leq \ell
  .\end{align*}
\item
  \(\gamma:\) Set \(e_i(1) \coloneqq 0\) to kill constants, and
  inductively on \(s\),
  \begin{align*}
  e_i(v_j \otimes a) = \delta_{ij}\alpha {}^{ \vee }_{i}(a) + v_j\otimes e_i(a) && a\in T^{s-1}(V), 1\leq j \leq \ell
  .\end{align*}
\end{enumerate}

One should show that these define a representation by checking the Serre
relations. Consider instead how this works in the
\({\mathfrak{g}}= {\mathfrak{sl}}_2\) case:

\begin{example}[?]

For \({\mathfrak{sl}}_2\), take the realization
\(({\mathbb{C}}, \left\{{\alpha}\right\}, \left\{{ \alpha {}^{ \vee }}\right\} )\)
corresponds to the matrix \(A = (s)\). Here \(T(V) = {\mathbb{C}}[x]\),
and since there are no Serre relations,
\(\tilde {\mathfrak{g}}= {\mathfrak{g}}\). We have
\(e = [0,1; 0,0], f = [0,0; 1,0]\) which generate the positive/negative
unipotent parts respectively. Then
\(h = \left\{{ \operatorname{diag}(h, -h) }\right\}\). Checking the
action:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(\alpha: { \begin{bmatrix}  {0} & {0} \\  {1} & {0} \end{bmatrix} } p = xp\)
  which raises degree by 1. M
\item

  \begin{align*}
  \beta:
  h(1) = \lambda(h)1
  \implies
  h(xp) = -\alpha(h)xp + x(hp)
  ,\end{align*}
  where \(p\in {\mathbb{C}}[x]_{g-1}\). For example,
  \begin{align*}
    h(x) &= -\alpha(x) + x \lambda(h) = (\lambda - \alpha)(h)x \\
    h(x^2) &= (\lambda - 2\alpha)(h)x^2
    ,\end{align*}
  so this acts diagonally and preserves degree.
\item
  Check
  \begin{align*}
  {
  \begin{bmatrix}
    {0} & {1} 
  \\
    {0} & {0}
  \end{bmatrix}
  } \cdot (1) &= 0 \\
  {
  \begin{bmatrix}
    {0} & {1} 
  \\
    {0} & {0}
  \end{bmatrix}
  } \cdot (xp) &= p + x {
  \begin{bmatrix}
    {0} & {1} 
  \\
    {0} & {0}
  \end{bmatrix}
  } p
  .\end{align*}
\end{enumerate}

Check that \(ex = 1+0\) and \(ex^2 = x+x = 2x\), so \(e\) acts by
differentiation.

\end{example}

Note that \({\mathfrak{h}}\) forms a subalgebra since it's a
nondegenerate map. This follows from the fact that we get a
representation \(\rho_{\lambda}\) of \(\tilde{\mathfrak{g}}\) on
\(T(V)\), which for each \(h\) acts nontrivially on some \(T(V)\). So
use \(\rho_{\lambda}\) to deduce the theorem for
\(\tilde{\mathfrak{g}}\):
\begin{align*}
\left\{{ [ x, y] {~\mathrel{\Big|}~}x,y\in {\mathfrak{h}}= \left\langle{\left\{{e_i, f_i}\right\}_{i=1}^\ell}\right\rangle }\right\}  \subseteq 
\tilde {\mathfrak{n}}^- + {\mathfrak{h}}+ \tilde{\mathfrak{n}}
= {\mathfrak{g}}
,\end{align*}
we'll show this sum is direct.

Let \(u = n^- + h + n^+ = 0\), then in \(T(V)\) we have
\(u(1) = n^-(1) + {\left\langle {\lambda},~{h} \right\rangle}1\), which
forces \({\left\langle {\lambda},~{h} \right\rangle} = 0\) for all
\(\lambda\in{\mathfrak{h}} {}^{ \vee }\) and thus \(h = 0\). Use the
restriction \(\tilde{\mathfrak{g}}\to \tilde{\mathfrak{n}}\) to get a
map \(U(\tilde{\mathfrak{n}}^-)\to T(V)\) out of the enveloping algebra,
using that \(T(V)\) is an associative algebra. Using
\(f_i \mapsto v_i\), this is surjective and in fact an isomorphism.
Sending \({\mathfrak{n}}^- \mapsto {\mathfrak{n}}^-(1)\) yields
\(\tilde{\mathfrak{n}}^- \subseteq U(\tilde{\mathfrak{n}}^-) = T(V)\).
This yields \(n^- = 0\), making the sum direct.

We can write
\(\tilde{\mathfrak{n}}^- = F\left\langle{f_1,\cdots, f_\ell}\right\rangle\)\\
and
\(\tilde{\mathfrak{n}}= F\left\langle{e_1,\cdots, e_\ell}\right\rangle\)
and by the PBW theorem, \(\dim \tilde{\mathfrak{g}}_\alpha < \infty\).
This uses that the weight spaces for \(\tilde{\mathfrak{n}}^-\) are
contained in \(U(\tilde{\mathfrak{n}}^-)\). Note that there is a
\emph{Cartan involution}
\begin{align*}
\tilde\omega: \tilde{\mathfrak{g}}&{\circlearrowleft}\\
e_i &\mapsto -f_i \\
f_i &\mapsto -e_i \\
h &\mapsto -h
.\end{align*}

Now to prove the theorem for \({\mathfrak{g}}\) itself, write
\(\tilde {\mathfrak{r}}\coloneqq\ker( \tilde{\mathfrak{g}}\xrightarrow{\gamma} {\mathfrak{g}}_\alpha) {~\trianglelefteq~}\tilde{\mathfrak{g}}\).
This is an ideal, and thus \({\mathfrak{h}}{\hbox{-}}\)stable. We can
thus write
\begin{align*}
\tilde{\mathfrak{r}}= 
\qty{
\bigoplus_{\alpha\in Q^+ \setminus\left\{{0}\right\}} {\mathfrak{r}}_{-\alpha}
}
\oplus 
\tilde{\mathfrak{r}}_0
\oplus 
\qty{
\bigoplus_{\alpha\in Q^+ \setminus\left\{{0}\right\}} {\mathfrak{r}}_{\alpha}
}
\end{align*}
where
\(\tilde{\mathfrak{r}}_{\beta} \coloneqq\tilde{\mathfrak{r}}\cap\tilde{\mathfrak{g}}_{\beta}\)
and
\(\tilde {\mathfrak{r}}_0 = \tilde{\mathfrak{r}}\cap{\mathfrak{h}}\). We
have ideals
\(\tilde{\mathfrak{r}}^{\pm} {~\trianglelefteq~}\tilde{\mathfrak{n}}^{\pm}\),
which are respectively generated by
\begin{align*}
\left\{{e_{i, j} = (\operatorname{ad}e_i)^{1 - a_{i, j}} (e_{j}) {~\mathrel{\Big|}~}i\neq j }\right\}
&&
\left\{{f_{i, j} = (\operatorname{ad}f_i)^{1 - a_{i, j}} (f_{j}) {~\mathrel{\Big|}~}i\neq j }\right\}
,\end{align*}
where \(\operatorname{ad}f_k (e_{i, j}) = 0\) for all \(k\) and
\(i\neq j\). Skipping a few things that are spelled out in the book,
e.g.~that \(\tilde {\mathfrak{r}}_0 = 0\), we conclude that
\(\tilde{\mathfrak{r}}= \tilde{\mathfrak{r}}^+ \oplus \tilde{\mathfrak{r}}^-\),
both of which are ideals in \(\tilde{\mathfrak{g}}\). Since
\(\tilde{\mathfrak{r}}_0 = 0\), we get
\({\mathfrak{h}}\subseteq {\mathfrak{g}}\), and using that \(\gamma\) is
surjective we have an isomorphism of \({\mathbb{C}}{\hbox{-}}\)modules
\begin{align*}
{\mathfrak{g}}= \tilde{\mathfrak{g}}/\tilde{\mathfrak{r}}= \tilde{\mathfrak{n}}^-/ \tilde{\mathfrak{r}}^- \oplus {\mathfrak{h}}\oplus \tilde{\mathfrak{n}}/ \tilde{\mathfrak{r}}^+
.\end{align*}

Write
\(\Delta \coloneqq\left\{{\alpha\in Q\setminus\left\{{0}\right\} {~\mathrel{\Big|}~}{\mathfrak{g}}_\alpha \neq 0}\right\}\)
the \emph{set of roots} and \({\mathfrak{g}}_{\alpha}\) the \emph{root
space}, then set
\begin{align*}
\Delta^+ &\coloneqq\Delta \cap Q^+ \\
\Delta^- &\coloneqq\Delta \cap(-Q^+) \\
\Delta &\coloneqq\Delta^+ \cup\Delta^-
.\end{align*}
Also for \(Y \subseteq \left\{{1,\cdots,\ell}\right\}\) write
\begin{align*}
\Delta_Y &\coloneqq\Delta \cap\qty{\bigoplus_{i\in Y} {\mathbb{Z}}\alpha_i } \\
{\mathfrak{g}}_Y &\coloneqq{\mathfrak{h}}\oplus \qty{ \bigoplus_{\alpha\in \Delta_Y} {\mathfrak{g}}_\alpha  }
.\end{align*}
We say \(Y\) is \emph{finite type} if \({\mathfrak{g}}_Y\) is finite
dimensional, and given \(A\) we can associate some matrix
\((a_{i, j})_{i, j\in Y}\).

\end{proof}

\begin{remark}

See Ch. 13 for how this generalizes the semisimple case.

\end{remark}

\hypertarget{weyl-groups-1.3-wednesday-september-01}{%
\section{Weyl Groups, 1.3 (Wednesday, September
01)}\label{weyl-groups-1.3-wednesday-september-01}}

\begin{remark}

We'll spend a few days discussing Weyl groups, since they're important
in the study of Schubert varieties. For other references, see

\begin{itemize}
\tightlist
\item
  Björner and Brenti: \emph{Combinatorics of Coxeter Groups}
\end{itemize}

\end{remark}

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

\begin{remark}

Recall that given a generalized Cartan matrix \(A\), there is an
associated realization
\(({\mathfrak{h}}, \pi \subseteq {\mathfrak{h}} {}^{ \vee }, \pi {}^{ \vee })\).

\end{remark}

\begin{definition}[Reflections]

For \(1\leq i \leq \ell\), define a \textbf{reflection}
\(s_i\in \mathop{\mathrm{Aut}}({\mathfrak{h}} {}^{ \vee })\) as
\begin{align*}
s_i(\chi) \coloneqq\chi - {\left\langle {\chi},~{\alpha_i {}^{ \vee }} \right\rangle}\alpha_i && \forall \chi \in {\mathfrak{h}} {}^{ \vee }
.\end{align*}

\end{definition}

\begin{remark}

One can check that this fixes a hyperplane, and
\(s_i^2 = \operatorname{id}\).

\end{remark}

\begin{definition}[Crystallographic Root Systems]

A subset \(\Phi\) of Euclidean space
\((V, {\left\langle {{-}},~{{-}} \right\rangle})\) is a
\textbf{crystallographic root system} in \(V\) iff

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\Phi\) is finite, \({\operatorname{span}}_{\mathbb{R}}\Phi = V\),
  and \(0\not\in\Phi\).
\item
  If \(\alpha \in \Phi\), then
  \({\mathbb{R}}\alpha \cap\Phi = \pm \alpha\).
\item
  If \(\alpha \in \Phi\), then \(s_\alpha\) leaves \(\Phi\) invariant
\item
  If \(\alpha, \beta\in \Phi\), then
  \({(\beta, \alpha) \over 2(\alpha, \alpha)} \in {\mathbb{Z}}\).
\end{enumerate}

\end{definition}

\begin{remark}

Note that for a Kac Moody Lie algebra, \(\Phi\) is often infinite, so
condition 1 can fail. Condition 2 can fail if \(\alpha\) is imaginary,
in which case \(n\alpha \in \Phi\) for some \(n\in {\mathbb{Z}}\).

\end{remark}

\begin{definition}[Weyl Groups]

Let \(W \subseteq \mathop{\mathrm{Aut}}({\mathfrak{h}} {}^{ \vee })\) be
the subgroup generated by
\(\left\{{s_i {~\mathrel{\Big|}~}1\leq i\leq \ell}\right\}\), then \(W\)
is said to be the \textbf{Weyl group} of \({\mathfrak{g}}\).

\end{definition}

\begin{definition}[Lengths]

Let \({\mathcal{W}}\) be the group generated by a fixed set \(S\) of
elements of order 2 in \(W\). Then for \(w\in {\mathcal{W}}\), the
\textbf{length} \(\ell(w)\) is the smallest number \(\ell\) such that
\(w = \prod_{i=1}^\ell s_i\).

\end{definition}

\begin{remark}

Note that \(\ell(1) = 0\), and for \(Y \subseteq S\), we set
\({\mathcal{W}}_Y\) to be the subgroup generated by
\(\left\{{s{~\mathrel{\Big|}~}s\in Y}\right\}\). We'll prove that any
Weyl group is a Coxeter group, but for now \(W\) is a Weyl group and
\({\mathcal{W}}\) is a Coxeter group.

\end{remark}

\begin{theorem}[1.3.11]

Let \(({\mathcal{W}}, S)\) be as above, then TFAE:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  The Coxeter condition: \({\mathcal{W}}\) is a quotient of the free
  group \(\widehat{{\mathcal{W}}}\) generated by \(S\), modulo the
  following relations:
\end{enumerate}

\begin{itemize}
\tightlist
\item
  \(s^2= 1\) for all \(s\in S\).
\item
  \((st)^{m_{s, t}} = 1\) for all \(s\neq t\) in \(S\) and for some
  integers \(m_{t, s} = m_{s, t} \geq 2\) (or possibly \(\infty\)).
\end{itemize}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  The root system condition: There exists a representation \(V\) of
  \({\mathcal{W}}\) over \({\mathbb{R}}\) together with a subset
  \(\Delta \subseteq V\setminus\left\{{0}\right\}\) such that
\end{enumerate}

\begin{itemize}
\item
  Symmetric: \(\Delta = -\Delta\)
\item
  \({\mathcal{W}}{\hbox{-}}\)invariance/stability: there exists a subset
  \(\pi \coloneqq\left\{{\alpha_s}\right\}_{s\in S} \subseteq \Delta\)
  such that for any \(\alpha \in \Delta\) exactly one of \(\alpha\) or
  \(-\alpha\) belongs to the set of positive linear combinations of
  ``simple roots'' \(\sum_{s\in S} {\mathbb{R}}_{>0} \alpha_s\). If
  \(\alpha\) is in this subset, we'll say \(\alpha\) is
  \textbf{positive}, and if \(-\alpha\) is in it, we'll say \(\alpha\)
  is \textbf{negative}.
\item
  For every \(s\in S\), if \(\alpha \neq \alpha_s\) and \(\alpha > 0\)
  is positive, then \(s\alpha_s <0\) is negative and \(s\alpha > 0\).
  \footnote{So the simple reflection changes the sign of only the
    corresponding simple root, and preserves the sign of all other
    simple roots.}
\item
  For \(s, t\in S\) and \(w\in {\mathcal{W}}\), then
  \(w \alpha_s = \alpha_t\) implies that \(wsw^{-1}= t\), so the group
  action is captured in a conjugation action.
\end{itemize}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{2}
\item
  The \emph{strong exchange} condition: For \(s\in S\) and
  \(v, w\in {\mathcal{W}}\) with \(\ell(vsv^{-1}w) \leq \ell(w)\), for
  any expression \(w = \prod_{i=1}^n s_i\) with \(s_i \in S\), we have
  \(vsv^{-1}w = \prod_{i\neq j}^n s_i\) for some \(j\).
\item
  The \emph{exchange condition}: For \(s\in S, w\in {\mathcal{W}}\) with
  \(\ell(sw) \leq \ell(w)\), then for any reduced expression
  \(w = \prod_{i=1}^n s_i\), we have \(sw = \prod_{i\neq j}^n s_i\) for
  some \(j\).
\end{enumerate}

\end{theorem}

\begin{remark}

These conditions show up in a lot of proofs!

\end{remark}

\begin{definition}[Crystallographic Coxeter groups]

If \(S\) is finite (which it will be for us), we can take \(V\) to be
finite dimensional. Writing
\(S \coloneqq\left\{{s_1, \cdots, s_\ell}\right\}\) and set
\(m_{ij} \coloneqq{\operatorname{Ord}}(s_i s_j)\). If every \(m_{ij}\)
is one of \(\left\{{2,3,4,6,\infty}\right\}\), call the Coxeter group
\textbf{crystallographic}.

\end{definition}

\begin{remark}

An open problem is that all Coxeter groups \emph{should} come from
geometry, e.g.~from projective varieties (?), but it's not clear what
these varieties should be. The crystallographic ones will precisely come
from Kac-Moody Lie algebras. This is closely related to problems
concerning KL polynomials: take an Ind variety, stratify it, and take
intersection cohomology.

\end{remark}

\begin{remark}

Every \emph{finite} irreducible Coxeter group (with exceptions
\(H_3, H_4, I_2(m)\)) occur as Weyl groups of crystallographic root
systems.

\end{remark}

\begin{proof}[of theorem, $1\implies 2$]

Let \(V\) be the \({\mathbb{R}}{\hbox{-}}\)module with basis
\(\left\{{\alpha_s {~\mathrel{\Big|}~}s\in S}\right\}\). For any
\(s\in S\), define an inner product by extending the following
\({\mathbb{R}}{\hbox{-}}\)bilinearly:
\begin{align*}
(\alpha_s, \alpha_s) &= 1 \\
(\alpha_{s_1}, \alpha_{s_2}) &= \cos({\pi \over m_{s_1s_2}}) && s_1\neq s_2 
.\end{align*}

For \(s, v\in V\), define
\begin{align*}
s(v) \coloneqq v - 2(v, \alpha_s) \alpha_s
.\end{align*}
A quick computation shows
\begin{align*}
s( \alpha_s) = \alpha_s - 2 (\alpha_s, \alpha_s) \alpha_s = - \alpha_s
.\end{align*}
One can check that the formula is \({\mathbb{R}}{\hbox{-}}\)linear, and
using this we have
\begin{align*}
s^2(v) 
&= s(v - 2(v, \alpha_s) \alpha_s) \\
&= s(v) - 2(v, \alpha_s) s(\alpha_s) \\
&= (v -2(v, \alpha_s) \alpha_s) - 2(v, \alpha_s)s( \alpha_s) \\
&= (v -2(v, \alpha_s) \alpha_s) - 2(v, \alpha_s)(- \alpha_s) \\
&= v
,\end{align*}
so \(s^2 = \operatorname{id}\). By assumption, we have
\((s_1 s_2)^{m_{s_1 s_2}}(v) = v\). Using that this formula factors
through the relations, we can extend this to an action
\({\mathcal{W}}\curvearrowright V\). Then
\begin{align*}
(s(v), s(v')) 
&= (v - 2(v, \alpha_s) \alpha_2, v' - 2(v', \alpha_s) \alpha_s) \\
&= (v, v') - 2(v', \alpha_s)(v, \alpha_s) - 2(v, \alpha_s)(\alpha_s, v') + 4(v, \alpha_s)(v', \alpha_s)(\alpha_s, \alpha_s) \\
&= (v, v') - 4(v', \alpha_s)(v, \alpha_s) + 4(v, \alpha_s)(v', \alpha_s)\\
&= (v, v')
,\end{align*}
where we've used that \(({-}, {-})\) is symmetric. Thus
\((wv, wv') = (v, v')\).

Let
\(\Delta\coloneqq\displaystyle\bigcup_{s\in S} \mathcal{W} ( \alpha_s)\),
we'll work with this more next time.

\end{proof}

\hypertarget{wednesday-september-08}{%
\section{Wednesday, September 08}\label{wednesday-september-08}}

\begin{remark}

Today: finish chapter one.

\end{remark}

\begin{definition}[Bruhat-Chevalley Partial Order]

For \(v, w\in W\) we set \(v\leq w \iff\) there exists
\({ {t}_1, {t}_2, \cdots, {t}_{p}} \in T\) such that

\begin{itemize}
\tightlist
\item
  \(v = t_p\cdots t_1 w\)
\item
  \(\ell(t_{j} \cdots t_1 w) \leq \ell(t_{j-1} \cdots t_1 w)\).
\end{itemize}

\end{definition}

\begin{definition}[Minimal length representatives]

For \(Y \subseteq S\) we set
\begin{align*}
W'_Y \coloneqq\left\{{\ell(wv) \geq \ell(w) \forall v\in W_Y}\right\}
.\end{align*}

\end{definition}

\begin{example}[?]

Consider \(W = S_3\) with
\(S\coloneqq\left\{{s_1, s_2}\right\} = \left\{{1, 2}\right\}\). The
Hasse diagram is the following:

\begin{center}
\begin{tikzcd}
    & {(3,2,1)} \\
    {(2,3,1)} && {(3,1,2)} \\
    {s_1 = (2,1,3)} && {(1,3,2) = s_2} \\
    & {(1,2,3)}
    \arrow[from=2-1, to=1-2]
    \arrow[from=2-3, to=1-2]
    \arrow[from=3-1, to=2-1]
    \arrow[from=3-3, to=2-3]
    \arrow[from=3-1, to=2-3]
    \arrow[from=3-3, to=2-1]
    \arrow[from=4-2, to=3-1]
    \arrow[from=4-2, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMSwwLCIoMywyLDEpIl0sWzAsMSwiKDIsMywxKSJdLFsyLDEsIigzLDEsMikiXSxbMCwyLCJzXzEgPSAoMiwxLDMpIl0sWzIsMiwiKDEsMywyKSA9IHNfMiJdLFsxLDMsIigxLDIsMykiXSxbMSwwXSxbMiwwXSxbMywxXSxbNCwyXSxbMywyXSxbNCwxXSxbNSwzXSxbNSw0XV0=}{Link
to Diagram}
\end{quote}

We have

\begin{itemize}
\item
  \(\emptyset \subseteq \left\{{1}\right\} \subseteq {1, 2}\)
\item

  \begin{align*}
  G/B = \left\{{0 \subseteq F^1 \subseteq F^2 \subseteq {\mathbb{C}}^3}\right\} \to \left\{{0 \subseteq F^2 \subseteq {\mathbb{C}}^3}\right\} \coloneqq{\operatorname{Gr}}_2({\mathbb{C}}^3) \to \left\{{0 \subseteq {\mathbb{C}}^3}\right\} = G/G
  .\end{align*}
  Note that Kumar writes
  \begin{align*}
  X^{\emptyset} &\coloneqq G/B \\
  X^{\left\{{1}\right\}} &= {\operatorname{Gr}}_2({\mathbb{C}}^3)\\
  X^{\left\{{1,2}\right\}} &= G/G
  .\end{align*}
\item
  For \(Y\coloneqq\left\{{1}\right\}\), we just have to check how
  lengths change upon swapping the first two positions. Thus
  \(W_Y = \left\{{e, s_1}\right\}\) since \((2,3,1)\) is minimal length.
  Similarly \((1,3,2)\) and \((1,2,3)\) are minimal length.
\end{itemize}

\includegraphics{figures/2021-09-08_14-05-35.png}

\begin{itemize}
\tightlist
\item
  For \(Y = \left\{{1, 2}\right\}\), we get \(W_Y = W\) with a minimal
  element \((1,2,3)\).
\end{itemize}

\end{example}

\begin{lemma}[?]

Fix a reduced expression \(w = \prod_{i\leq n} s_i\). Then \(v\leq w\)
iff there exist indices \(1\leq j_1 < j_2 < \cdots < j_p \leq n\) such
that \(v = \prod_{i\neq j_k} s_i\).

\end{lemma}

\begin{example}[?]

For \(m_{12} = 3\), if \((s_1 s_2)^{m_{12} = 3} = e\), so
\(s_1 s_2 s_1 = s_2 s_1 s_2\), which is a braid relation that
corresponds to \((3,2,1)\). Let \(w_0\) be the maximal element (which
generally only works when the Coxeter group is finite), so here
\(w_0 = s_1 s_2 s_1\). We can cross out various reflections to get
closure relations:

\begin{center}
\begin{tikzcd}
    & {w_0 = s_1 s_2 s_1 = s_2 s_1 s_2} \\
    {s_1 s_2} && {s_2 s_1} \\
    {s_1} && {s_2} \\
    & e
    \arrow[from=4-2, to=3-1]
    \arrow[from=4-2, to=3-3]
    \arrow[from=3-3, to=2-3]
    \arrow[from=2-3, to=1-2]
    \arrow[from=2-1, to=1-2]
    \arrow[from=3-1, to=2-1]
    \arrow[from=3-1, to=2-3]
    \arrow[from=3-3, to=2-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMSwwLCJ3XzAgPSBzXzEgc18yIHNfMSA9IHNfMiBzXzEgc18yIl0sWzAsMSwic18xIHNfMiJdLFsyLDEsInNfMiBzXzEiXSxbMCwyLCJzXzEiXSxbMiwyLCJzXzIiXSxbMSwzLCJlIl0sWzUsM10sWzUsNF0sWzQsMl0sWzIsMF0sWzEsMF0sWzMsMV0sWzMsMl0sWzQsMV1d}{Link
to Diagram}
\end{quote}

Here for \(Y = \left\{{1}\right\} = \left\{{s_1}\right\}\), we get
minimal length elements \(e, s_2, s_1 s_2\).

\end{example}

\begin{example}[?]

In general, we start with a GCM \(A\), take a realization
\(({\mathfrak{h}}, \pi, \pi {}^{ \vee })\), get Kac-Moody Lie algebra
\({\mathfrak{g}}\), and extract a group \(W\) which we now know is a
Coxeter group. Write
\(\left\{{{ {\alpha}_1, {\alpha}_2, \cdots, {\alpha}_{\ell}}}\right\} \subseteq {\mathfrak{h}} {}^{ \vee }\)
and \(S = \left\{{s_1, \cdots, s_\ell}\right\}\), then for any
\(1\leq i\leq \ell\) set
\begin{align*}
s_i(\chi) \coloneqq\chi - {\left\langle {\chi},~{\alpha_i {}^{ \vee }} \right\rangle}\alpha_i && \forall \chi\in{\mathfrak{h}} {}^{ \vee }
.\end{align*}
Fix a real form \({\mathfrak{h}}_{\mathbb{R}}\) of \({\mathfrak{h}}\)
satisfying

\begin{itemize}
\tightlist
\item
  \(\pi {}^{ \vee }\subseteq {\mathfrak{h}}_{\mathbb{R}}\),
\item
  \(\alpha_i({\mathfrak{h}}_{\mathbb{R}}) \subseteq {\mathbb{R}}\) for
  all \(1\leq i \leq \ell\).
\end{itemize}

\end{example}

\begin{definition}[Dominant Chamber]

Define the \textbf{dominant chamber}
\(D_{\mathbb{R}}\subseteq {\mathfrak{h}}_{\mathbb{R}} {}^{ \vee }\coloneqq\operatorname{Hom}_{{\mathsf{{\mathbb{R}}}{\hbox{-}}\mathsf{Mod}}}({\mathfrak{h}}_{\mathbb{R}}, {\mathbb{R}})\)
as
\begin{align*}
D_{\mathbb{R}}\coloneqq\left\{{ \lambda\in {\mathfrak{h}}_{\mathbb{R}} {}^{ \vee }{~\mathrel{\Big|}~}\lambda(\alpha_i) \geq 0 \, \forall i}\right\}
.\end{align*}

\end{definition}

\begin{definition}[Tits Cone]

Define the \textbf{Tits cone} as
\begin{align*}
C \coloneqq\displaystyle\bigcup_{w\in W} wD_{\mathbb{R}}
.\end{align*}

\end{definition}

\begin{remark}

Consider the reductive group \({\mathsf{Sp}}_4({\mathbb{C}})\), which is
semisimple, simply connected, and connected. One way to realize this
group is as
\begin{align*}
{\mathsf{Sp}}_4({\mathbb{C}}) \coloneqq\left\{{ g\in \operatorname{GL}_4({\mathbb{C}}) {~\mathrel{\Big|}~}\Theta(g) = g}\right\}
\end{align*}
for \(\Theta\) some involution of \(\operatorname{GL}_4({\mathbb{C}})\).
Noting that we always have associated root datum
\((n, \left\{{ \alpha_i}\right\}_{i=1}^\ell ,\left\{{\alpha_i {}^{ \vee }}\right\}_{i=1}^\ell )\),
here we have
\begin{align*}
{\mathsf{Sp}}_4({\mathbb{C}}) = (2, \left\{{ (1,-1), (0, 2)}\right\}, \left\{{(1, -1), (0, 1)}\right\} )
.\end{align*}
This yields a GCM
\begin{align*}
A = {
\begin{bmatrix}
  {2} & {-2} 
\\
  {-1} & {2}
\end{bmatrix}
}
,\end{align*}
which comes from computing
\((A)_{ij} \coloneqq\alpha_i(\alpha_j {}^{ \vee })\). Here
\begin{align*}
G/Z(G) = (2, \left\{{ (1, 0), (0, 1) }\right\} , \left\{{ (2, -2), (-1, 2) }\right\} )
.\end{align*}
Note that these two root data are distinct over \({\mathbb{Z}}\). We can
consider the real form \({\mathfrak{h}}_{\mathbb{R}} {}^{ \vee }\):

\includegraphics{figures/2021-09-08_14-30-01.png}

We have

\begin{itemize}
\tightlist
\item
  \(\chi \in {\mathfrak{h}}_{\mathbb{R}} {}^{ \vee }= \left\{{(x, y)}\right\}\),
\item
  \(s_1(x, y) = (x, y) - {\left\langle {(x, y)},~{(1, -1)} \right\rangle}(1, -1) = (y, x)\)
\item
  \(s_2(x, y) = (x, y) - {\left\langle {(x, y)},~{(0, 1)} \right\rangle}(0, 2) = (x, -y)\)
\end{itemize}

We can look at the \(W{\hbox{-}}\)orbits of these, and it turns out to
recover all of the roots:

\includegraphics{figures/2021-09-08_14-35-37.png}

\(W \subseteq \mathop{\mathrm{Aut}}({\mathfrak{h}}_{\mathbb{R}} {}^{ \vee })\)
is the subgroup given by \(\left\{{s_1, s_2}\right\}\), and there are
maps
\begin{align*}
s_1 s_2: (x, y) &\mapsto (-y, x)\\
s_2 s_1 s_2: (x, y) &\mapsto (y, -x)\\
s_1s_2 s_1 s_2: (x, y) &\mapsto (-x, -y)\\
s_2 s_1s_2 s_1 s_2: (x, y) &\mapsto (-x, y)\\
\vdots &\quad  \vdots \\
(s_1s_2)^4: (x, y) &\mapsto (x, y) \implies m_{12} = 4
.\end{align*}
Here we've used that \((s_1 s_2)^2 = (s_2 s_1)^2\). We can then find the
dominant chamber:

\includegraphics{figures/2021-09-08_14-40-17.png}

For \(\lambda \in D_{\mathbb{R}}\), we set
\(W_\lambda \coloneqq\left\{{w\in W {~\mathrel{\Big|}~}w(\lambda) = \lambda}\right\}\).
This is generated by the simple reflections it contains. Setting
\(Y = Y(\lambda) = \left\{{s_i \in S {~\mathrel{\Big|}~}\lambda( \alpha_i {}^{ \vee }) = 0}\right\}\),
we actually get \(W_\lambda = W_Y\).

\end{remark}

\begin{remark}

Recall what regular weights are!

\end{remark}

\hypertarget{category-mathcalo-friday-september-10}{%
\section{\texorpdfstring{Category \({\mathcal{O}}\) (Friday, September
10)}{Category \{\textbackslash mathcal\{O\}\} (Friday, September 10)}}\label{category-mathcalo-friday-september-10}}

Counterexamples: Kac Moodys that aren't usual Lie algebras: affine Kac
Moodys.

\begin{remark}

Our setup: \(A \leadsto ({\mathfrak{h}},\pi,\pi {}^{ \vee })\). Fix
\(\lambda \in {\mathfrak{h}} {}^{ \vee }\) and
\(c\in {\mathbb{C}}_{ \lambda} \ni z\) a representation of
\({\mathfrak{h}}\) by \(x.\cdot \coloneqq\lambda(x)z\). Recall that we
have a triangular decomposition
\({\mathfrak{g}}= {\mathfrak{n}}^- \oplus {\mathfrak{h}}\oplus {\mathfrak{n}}\)
with \({\mathfrak{h}}\oplus {\mathfrak{n}}\leq {\mathfrak{b}}\) a
subalgebra of the Borel. Since
\({\mathfrak{n}}{~\trianglelefteq~}{\mathfrak{b}}\) is an ideal, we can
quotient to extend the representation
\begin{align*}
{\mathfrak{b}}\to {\mathfrak{b}}/{\mathfrak{n}}\cong {\mathfrak{h}}\xrightarrow{\lambda} {\mathbb{C}}_{\lambda}
.\end{align*}
This extends from \({\mathfrak{h}}\) to \({\mathfrak{b}}\) by making it
zero on \({\mathfrak{n}}\), and generally one can do this with
nilradicals.

\end{remark}

\begin{definition}[Verma Modules]

\begin{align*}
M(\lambda) \coloneqq U({\mathfrak{g}}) \otimes_{U({\mathfrak{b}})} {\mathbb{C}}_{\lambda} \in {\mathsf{{\mathfrak{g}}}{\hbox{-}}\mathsf{Mod}}
,\end{align*}
where \({\mathfrak{b}}\curvearrowright{\mathbb{C}}_{\lambda}\) extends
to the universal enveloping algebra.

\end{definition}

\begin{remark}

The PBW theorem implies that every
\(M(\lambda)\cong U({\mathfrak{n}}^-) \otimes_{\mathbb{C}}{\mathbb{C}}_{\lambda}\)
as vector spaces, which is in fact an isomorphism in
\({\mathsf{{\mathfrak{b}}^-}{\hbox{-}}\mathsf{Mod}}\). This means
\(M( \lambda)\) is a weight module for \({\mathfrak{h}}\), i.e.~there is
a decomposition
\(M(\lambda) = \bigoplus_{\mu \in {\mathfrak{h}} {}^{ \vee }}M(\lambda)_{ \mu}\)
where
\begin{align*}
M(\lambda)_{\mu} \coloneqq\left\{{v\in M( \lambda) {~\mathrel{\Big|}~}h\cdot v = \mu(h) v,\,\quad h\in {\mathfrak{h}}}\right\}
.\end{align*}

\end{remark}

\begin{definition}[Highest weight modules]

Any nonzero quotient \(L\) of \(M(\lambda)\) in
\({\mathsf{{\mathfrak{g}}}{\hbox{-}}\mathsf{Mod}}\) is a \textbf{highest
weight module} with highest weight \(\lambda\).

\end{definition}

\begin{remark}

Why \emph{highest weight}? There is a partial order on weights:
\begin{align*}
\mu \leq \lambda\iff \lambda- \mu\in Q^+ \coloneqq{\mathbb{Z}}_{> 0} \pi
.\end{align*}
Also note that \(M( \lambda)\) is a highest weight module.

\end{remark}

\begin{definition}[Category $\OO$]

There is a full subcategory
\({\mathcal{O}}\leq {\mathsf{{\mathfrak{g}}}{\hbox{-}}\mathsf{Mod}}\)
where every \(M\in {\operatorname{Ob}}({\mathcal{O}})\) satisfies the
following:

\begin{itemize}
\tightlist
\item
  (Finite multiplicities) \(M\) is a weight module with
  finite-dimensional weight spaces.
\item
  There exist finitely many weights
  \({ {\lambda}_1, {\lambda}_2, \cdots, {\lambda}_{k}}\in {\mathfrak{h}} {}^{ \vee }\)
  (depending on \(M\)) such that
  \(P(M) \subseteq \displaystyle\bigcup_{1\leq j\leq k} {\mathfrak{h}} {}^{ \vee }_{\leq \lambda_j}\):
\end{itemize}

\includegraphics{figures/2021-09-10_14-08-34.png}

\end{definition}

\begin{lemma}[?]

Any \(M(\lambda)\) has a unique proper maximal
\({\mathfrak{g}}{\hbox{-}}\)submodule \(M'(\lambda)\). In particular,
\(\lambda\not \in M'(\lambda)\), and there is a unique irreducible
quotient \(L(\lambda) \coloneqq M( \lambda)/M'( \lambda)\).

\begin{quote}
The proof is easy: use that \(\lambda\) generated \(M(\lambda)\) as a
\({\mathfrak{g}}{\hbox{-}}\)module.
\end{quote}

\end{lemma}

\begin{lemma}[?]

For any irreducible \(L \in {\operatorname{Ob}}({\mathcal{O}})\), there
exists a unique \(\lambda \in {\mathfrak{h}} {}^{ \vee }\) such that
\(L \cong L(\lambda)\).

\end{lemma}

\begin{definition}[Dominant Integral Weights]

Define the \textbf{dominant integral weights}
\begin{align*}
D \coloneqq\left\{{ \lambda\in {\mathfrak{h}} {}^{ \vee }{~\mathrel{\Big|}~}\forall \alpha_i {}^{ \vee }\in \pi {}^{ \vee },\,\, {\left\langle { \lambda},~{\alpha_i {}^{ \vee }} \right\rangle} \in {\mathbb{Z}}_{> 0} }\right\}
.\end{align*}

\end{definition}

\begin{definition}[Maximal integrable highest weight modules]

For \(\lambda \in D\), define \(M_1(\lambda) \subseteq M( \lambda)\) as
the submodule generated by
\(\left\{{ f_i^{ \lambda(\alpha_i {}^{ \vee }) + 1 } \otimes 1}\right\}_{i=1}^\ell\),
and define
\begin{align*}
L^{\max}( \lambda) \coloneqq{M(\lambda) \over M_1( \lambda)}
,\end{align*}
the operators that act locally nilpotently (so there is an exponent
depending on the vector)

\end{definition}

\begin{example}[?]

Let \(A = {\left[ {2} \right]}\) be a \(1\times 1\) GCM, which yields
\(({\mathbb{C}}, \left\{{ 2}\right\}, \left\{{ 1}\right\})\leadsto {\mathfrak{sl}}_2({\mathbb{C}})\).
Given \(\lambda\in {\mathbb{C}},\) we have
\begin{align*} 
M( \lambda) 
&= U({\mathfrak{sl}}_2) \otimes_{U({\mathfrak{b}})} {\mathbb{C}}_{\lambda} \\
&\cong U({\mathfrak{n}}^-) \otimes_{\mathbb{C}}{\mathbb{C}}_{\lambda} \\
&= {\mathbb{C}}[y] \otimes_{\mathbb{C}}{\mathbb{C}}_{ \lambda}
.\end{align*}
where noting that \({\mathfrak{n}}^- = \left\langle{f_i}\right\rangle\)
and \({\mathfrak{n}}= \left\langle{e_i}\right\rangle\), we identify the
variable \(y\) with \(f\).

This has weights \(\lambda, \lambda-2, \lambda-4, \cdots\), identifying
elements as \(y^k \otimes 1\). How do \(e,f,h \in {\mathfrak{g}}\) act
in this basis?

\begin{itemize}
\tightlist
\item
  \(h(y^k \otimes 1) = (hy^k) \otimes 1 = (\lambda - 2k)(y^k\otimes 1)\).
\item
  \(f(y^k\otimes 1) = y(y^k\otimes 1) = y^{k+1}\otimes 1\).
\item
  \(e\): more complicated!
\end{itemize}

The game: move \(e\)s across the tensor product to kill terms:

\begin{itemize}
\item
  For \(k=0\):
  \begin{align*}
  e(1\otimes 1) = e\otimes 1 = 1\otimes e(1) = 0
  \end{align*}
  since we extended \(\lambda\) by zero on \({\mathfrak{n}}\).
\item
  For \(k=1\):
  \begin{align*}
  e(y\otimes 1) 
  &= e(f\otimes 1) \\
  &= (ef)\otimes 1 \\
  &= ([ef] + fe)\otimes 1 \\
  &= [ef] \otimes 1 \\
  &= \alpha {}^{ \vee }\otimes 1 \\
  &= 1\otimes\alpha {}^{ \vee }\cdot 1 \\
  &= \lambda( \alpha {}^{ \vee })(1\otimes 1) \\
  &= \lambda
  ,\end{align*}
  using \(ef = [e,f] + fe = ef-fe + fe\) and
  \(fe\otimes 1 = f\otimes e(1) = 0\).
\item
  For \(k=2\):
  \begin{align*}
  eff\otimes 1 
  &= ([ef] + fe)f \otimes 1 \\
  &=( \alpha {}^{ \vee }f + fef )\otimes 1 \\
  &= ( \alpha {}^{ \vee }f + f([ef] + fe) ) \otimes 1 \\
  &= (\alpha {}^{ \vee }f + f[ef]) \otimes 1 \\
  &= (\alpha {}^{ \vee }f + f \alpha {}^{ \vee }) \otimes 1 && f \alpha {}^{ \vee }\in {\mathfrak{h}}\\
  &= ( \alpha {}^{ \vee }f + \lambda f) \otimes 1 \\
  &= ( [\alpha {}^{ \vee }, f] + f \alpha {}^{ \vee }+ \lambda f) \otimes 1 \\ 
  &= ( - \alpha( \alpha {}^{ \vee }) f + 2 \lambda f) \otimes 1 && \text{using Kac-Moody relns.} \\
  &= 2( \lambda- 1)f \otimes 1
  .\end{align*}
\end{itemize}

Then general pattern is
\(e(y^k \otimes 1) = k ( \lambda - (k-1) ) \qty{ y^{k-1} \otimes 1 }\).

Here
\begin{align*}
D =\left\{{ \lambda\in {\mathfrak{h}} {}^{ \vee }= {\mathbb{C}}{~\mathrel{\Big|}~}{\left\langle { \lambda},~{ \alpha {}^{ \vee }} \right\rangle} \in {\mathbb{Z}}_{>0} }\right\} = {\mathbb{Z}}_{>0} \subseteq {\mathbb{C}}= {\mathfrak{h}} {}^{ \vee }
\end{align*}
and for \(\lambda \in D\),\\
\begin{align*}
M_1( \lambda) = \left\{{f^{ \lambda( \alpha_i {}^{ \vee }) + 1} \otimes 1}\right\}_{1\leq i \leq \ell = 1} = \left\{{f^{\lambda+ 1} \otimes 1}\right\}
.\end{align*}
Note that \(e\cdot f^{ \lambda+1}\otimes 1 = 0\), which can be checked
from the above formula:
\begin{align*}
e(y^{\lambda+1} \otimes 1) = ( \lambda+1)( \lambda- \lambda) y^{ \lambda} = 0
.\end{align*}
Thus
\(M_1( \lambda) = {\mathbb{C}}\left\langle{y^{\lambda+1}, y^{ \lambda+2}, \cdots}\right\rangle\).
Finally,
\begin{align*}
{M( \lambda) \over M_1( \lambda)} = L^{\max}( \lambda) = L( \lambda)
.\end{align*}

\end{example}

\hypertarget{tits-systems-5.1-monday-september-13}{%
\section{Tits Systems, 5.1 (Monday, September
13)}\label{tits-systems-5.1-monday-september-13}}

\begin{remark}

The basic setup from the book:
\begin{align*}
A\leadsto ({\mathfrak{h}}, \pi, \pi {}^{ \vee }) \leadsto {\mathfrak{g}}\leadsto (W, S)
.\end{align*}
We'll think of \(G\leadsto ({\mathfrak{h}}, \pi, \pi {}^{ \vee })\) as
the root data associated to a semisimple simply connected connected
algebraic group. Warning: this association isn't unique in the
non-semisimple case! Noting that \((W, S)\) is a Coxeter group, is there
a way to recover an algebra \({\mathfrak{g}}\) and a Kac-Moody group
\({\mathcal{G}}\)?

For today: take

\begin{itemize}
\item
  \(G\coloneqq\operatorname{GL}_n\), Note that \(G\) is not semisimple
  or simply connected.
\item
  \(B\) the fixed Borel (maximum connected closed solvable subgroup) of
  upper-triangular matrices. Flag varieties are homogeneous projective
  spaces, so \(G/B\) is a flag variety.
\item
  \(T\) the maximal torus of diagonal matrices
\item
  \(N = N_G(T)\) to be the subgroup generated by all permutation and
  scalar matrices.
\item
  The Weyl group \(W\coloneqq N/ B \cap N = N/T\) since
  \(B \cap N = T\). Note that \(W\cong S_n\) is a Coxeter group.
\item
  \(S \subseteq W\) is the subset of simple reflections, writing
  \(w = (w_1, \cdots, w_n)\) and taking only those permutations that
  transpose two adjacent coordinates, so
  \begin{align*}
  \tau_{k}: (w_1, \cdots, w_k, w_{k+1}, \cdots, w_n) &\mapsto
  (w_1, \cdots, w_{k+1}, w_{k}, \cdots, w_n)
  .\end{align*}

  This can be written as
  \(\left\langle{\tau_k}\right\rangle \coloneqq\left\langle{(k, k+1) {~\mathrel{\Big|}~}{1\leq k \leq n-1}}\right\rangle\).
\end{itemize}

\end{remark}

\begin{remark}

More generally, \(G \supseteq B \supseteq T\) and we set
\(W \coloneqq N_G(T) / Z_G(T)\) and show \(Z_G(T) = T\), but what is
\(B \cap N\) generally? Maybe use the fact that \(N_G(B) = B\)? Or that
the unipotent radical intersects it trivially.

\end{remark}

\begin{definition}[Tits Systems]

A \textbf{Tits system} is a tuple \((G,B,N,S)\) where \(B,N\leq G\) are
subgroups and \(S \subseteq W = N/B \cap N\), which collectively adhere
to the following axioms:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(B \cap N {~\trianglelefteq~}N\),
\item
  \(B, N\) generate \(G\),
\item
  For all \(s_i\in S\), we have \(sBs^{-1}\not\subseteq B\)
\item
  For \(w\in S_n\) and \(s\in S\), defining
  \(C(x) \coloneqq B \mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu B \subseteq G\)
  for any coset representative
  \(\mkern 1.5mu\overline{\mkern-1.5mux\mkern-1.5mu}\mkern 1.5mu\) of
  \(x\) in \(N\), we require \(C(s) C(w) \subseteq C(w) \cup C(sw)\).
\end{enumerate}

\end{definition}

\begin{remark}

Consider elements in \(BN\) for \(\operatorname{GL}_n\): \(B\) is upper
triangular, \(N\) has one (possibly) nonzero entry in each row/column,
and multiplying this can ``smear'' the entries upward by filling a
column above an entry:

\includegraphics{figures/2021-09-13_14-15-01.png}

Similarly, multiplying on the right smears rightward, and it's not so
hard to convince yourself that these generate \(\operatorname{GL}_n\).

For the conjugation axiom, consider the following:

\includegraphics{figures/2021-09-13_14-17-38.png}

We also have
\(B\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu B\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu B \subseteq B\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu B \cup B \mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu B\).
To prove this, we'll show

\begin{itemize}
\tightlist
\item
  \(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu B \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu \subseteq\)
  the RHS,
\item
  The right-hand side is stable under the \(B\times B\) action of
  left/right multiplication.
\end{itemize}

To see the first, consider the example:

\includegraphics{figures/2021-09-13_14-25-25.png}

For the second, consider
\begin{align*}
(1,3,2,4)(3,4,1,2) = (2,4,1,3)
.\end{align*}
The hard case is when lengths of the result change.

\end{remark}

\begin{definition}[Parabolics]

Any \(B \subseteq P \subseteq G\) is called a \textbf{standard
parabolic}. Any subgroup \(Q\) conjugate to \(P\) is called
\textbf{parabolic}.

\end{definition}

\begin{remark}

Standard parabolics correspond to subsets \(Y\) of simple reflections
\(\emptyset \subseteq Y \subseteq S\). Any subgroup containing the upper
triangular matrices looks like the following:

\includegraphics{figures/2021-09-13_14-32-29.png}

For \(P_Y\), we take everything but skip the first index.

\end{remark}

\begin{remark}

\envlist

\begin{itemize}
\item
  Take
  \(S \subseteq \left\{{w\in W {~\mathrel{\Big|}~}w^2 = \operatorname{id}}\right\}\)
  a subset of order 2 elements.
\item
  \(P_Y = BW_Y B = \displaystyle\coprod_{s\in U} B \mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu B \subseteq {\mathcal{G}}\).
\item
  \(G = \displaystyle\coprod_{w\in W} C(w)\)
\item
  There is a decomposition into double coset orbits:
  \begin{align*}
  G = \displaystyle\coprod_{w\in \dcoset{W_Y}{W}{W_{Y'}} } P_Y w P_{Y'}
  .\end{align*}
\item
  We have
  \begin{align*}
  C(s) C(w) = 
  \begin{cases}
  C(sw)  & \ell(sw) \geq \ell(w) 
  \\
  C(w) \cup C(s) & \ell(sw) = \ell(w).
  \end{cases}
  \end{align*}
\item
  \((W, S)\) is a Coxeter group.
\item
  For any parabolic \(P\) (not necessarily standard), its normalizer
  satisfies \(N_G(P) = P\). Note that you can plug in a Borel here.
  Moreover \(G/P = G/N_G(P)\), which parameterizes parabolic subgroups
  of \(G\).

  \begin{itemize}
  \item
    \(w\in W_Y'(Y) \cong W/W_Y\). Fixing a \emph{reduced} decomposition
    \(w = w_1\cdots w_k\), i.e.~\(\ell(w) =\sum_{i=1}^k \ell(w_i)\).
  \item
    For any \(A_i \subseteq C(w_i)\) where \(A_i \to C(w_i)/B\) is
    bijective (resp. surjective), the multiplication
    \(\phi: A_1\times \cdots A_{k} \to BwP_Y/P_Y\) is bijective (resp.
    surjective).
  \end{itemize}
\end{itemize}

\end{remark}

\hypertarget{generalized-flag-varieties-7.1-wednesday-september-15}{%
\section{Generalized Flag Varieties, 7.1 (Wednesday, September
15)}\label{generalized-flag-varieties-7.1-wednesday-september-15}}

\begin{remark}

Most of the things we'll look at will be motivated by the finite-type
case, but the statements still go through more generally. The setup:
\(A\) a GCM \(\leadsto\) root datum
\(({\mathfrak{h}}, \pi, \pi {}^{ \vee }) \leadsto {\mathfrak{g}}\) a
Kac-Moody Lie algebra \(\leadsto (W, S)\) a Coxeter group
\(\leadsto T \subseteq B\) a maximal torus, where
\(T = \operatorname{Hom}_{\mathbb{Z}}({\mathfrak{h}}_{\mathbb{Z}}, {\mathbb{C}}^{\times})\)
and \(B\) plays the role of the Borel, \(\leadsto {\mathcal{G}}\) a
Kac-Moody group. Here \({\mathfrak{h}}_{\mathbb{Z}}\) is the integer
span of coroots, using that \({\mathfrak{h}}\subseteq \pi {}^{ \vee }\).
Note that since \({\mathcal{G}}\) arises from a Tits system, so even
though we haven't described it set-theoretically yet, we know many nice
properties it has by previous propositions.

\end{remark}

\begin{fact}

For \(G\in {\mathsf{Alg}}{\mathsf{Grp}}\) arbitrary and \(H\leq G\), the
quotient space \(G/H\) is a variety (See Springer's book for a proof).
Write \(G/H = (X, a)\) where \(a= H/H\) is a distinguished point.
Quotients have a universal property: for any pair \((Y, b)\) of pointed
\(G{\hbox{-}}\)spaces whose isotropy (stabilizer) group contains \(H\),
there exists a unique equivariant pointed morphism \(\phi: G/H\to Y\)
such that \(\phi(a) = b\).

\end{fact}

\begin{remark}

Today: we defined a flag variety to be any projective homogeneous space,
and today we'll see that \(G/B\) is a \emph{projective} variety. In
fact, we'll show that \({\mathcal{G}}/P_Y\) is a projective
\emph{ind-variety}, where \(P_Y\) is the standard parabolic coming from
the Tits system.

\end{remark}

\begin{definition}[Ind-varieties]

An \textbf{Ind-variety} is a set with a countable filtration
\(X_0 \subseteq X_1 \subseteq \cdots\) such that

\begin{itemize}
\tightlist
\item
  \(X = \colim_n X_n = \displaystyle\bigcup_n X_n\),
\item
  Each \(X_n \hookrightarrow X_{n+1}\) is a closed embedding of
  finite-dimensional varieties.
\end{itemize}

\(X\) will be projective/affine iff its filtered pieces are
projective/affine.

\end{definition}

\begin{remark}

Note that we don't require a stratification here, but there will be a
stratification on the flag varieties we'll use, which induces a
filtration.

\end{remark}

\begin{example}[?]

Infinite affine space \({\mathbb{A}}^\infty_{/ {k}}\) can be written as
\begin{align*}
{\mathbb{A}}^\infty_{/ {k}}  = \left\{{ (a_1,a_2,\cdots) {~\mathrel{\Big|}~}a_i \in k, \text{ finitely many } a_i\neq 0}\right\}
.\end{align*}
The filtration is given by
\begin{align*}
{\mathbb{A}}^1 &\hookrightarrow{\mathbb{A}}^\infty \\
x &\mapsto (x,0,0,\cdots) \\ \\
{\mathbb{A}}^2 &\hookrightarrow{\mathbb{A}}^\infty \\
(x, y) &\mapsto (x,y,0,\cdots) \\
\vdots & 
\end{align*}

\end{example}

\begin{example}[?]

For \(V\in {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) with
\(\dim_{\mathbb{C}}V = \infty\), we have
\(V\cong {\mathbb{A}}^\infty_{/ {{\mathbb{C}}}}\) as Ind-varieties.

\end{example}

\begin{example}[?]

For any \(V\in {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\), the
space \({\mathbb{P}}(V) \coloneqq{\operatorname{Gr}}_1(V)\) (the space
of lines in \(V\)) is a projective Ind-variety.

\end{example}

\begin{remark}

For any integrable highest weight \({\mathfrak{g}}{\hbox{-}}\)module
\(V = V(\lambda)\) for \(\lambda \in D_{\mathbb{Z}}\) an integral
dominant weight, this will yield a \({\mathcal{G}}{\hbox{-}}\)module.
Here for \({\mathfrak{g}}\) semisimple, it integrates to the simply
connected \({\mathcal{G}}\).

\end{remark}

\begin{definition}[?]

For any \(v_\lambda \neq 0\in V\), define
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5mu\iota\mkern-1.5mu}\mkern 1.5mu_v: {\mathcal{G}}&\to {\mathbb{P}}(V) \\
g &\mapsto [gv_\lambda]
,\end{align*}

\end{definition}

\begin{definition}[?]

For any \(Y \subseteq \left\{{1,\cdots,\ell}\right\}\), define \(D_Y^0\)
the \textbf{\(Y{\hbox{-}}\)regular weights} by
\begin{align*}
D_Y^0 \coloneqq\left\{{ \lambda\in D_{\mathbb{Z}}{~\mathrel{\Big|}~}{\left\langle { \lambda},~{ \alpha_i} \right\rangle} = 0 \iff i \in Y}\right\}
.\end{align*}
This partitions the integral dominant chamber:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{41pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-09-15_14-18.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{definition}

\begin{lemma}[?]

For \(\lambda \in D_Y^0\) the map
\(\mkern 1.5mu\overline{\mkern-1.5mu\iota\mkern-1.5mu}\mkern 1.5mu_v\)
factors through \({\mathcal{G}}/P_Y\) to give an injection
\begin{align*}
\iota_v: G/P_Y \hookrightarrow{\mathbb{P}}(V)
.\end{align*}
So any Kac-Moody maps into an Ind-variety.

\end{lemma}

\begin{remark}

We'll show that \(\operatorname{im}\iota_v \subseteq {\mathbb{P}}(V)\)
is closed, i.e.~that its intersection with any finite filtered piece is
closed. The variety structure will be induced from this embedding.

\end{remark}

\begin{proof}[?]

We have a distinguished point \([v_\lambda] \in {\mathbb{P}}(V)\), so
\({\operatorname{Stab}}_G([v_ \lambda]) \supseteq P_Y\). Showing this
amounts to showing that for all \(s\in Y\),
\(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu\in G\)
fixes \([v _{\lambda} ]\), but this follows from the definition of
\(v_\lambda\).

\end{proof}

\begin{remark}

A great class of varieties: Bott-Samelson-Demazure-Hansen varieties,
which capture the geometry of words in Coxeter groups. We'll have
\(w\in W, \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu\in N\),
and we'll define \({\mathfrak{W}}\ni w\) as words:
\begin{align*}
{\mathfrak{W}}\coloneqq\left\{{w = (s_{i_1}, \cdots, s_{i_n} ) {~\mathrel{\Big|}~}n\geq 0}\right\}
,\end{align*}
which is a poset under deleting symbols. For any
\(w\in {\mathfrak{W}}\), define
\(Z_w \coloneqq\prod_{k\leq n} P_{i_k} / B{ {}^{ \scriptscriptstyle\times^{n} } }\),
where the action of the Borel is the \emph{right mixed space action}:
\begin{align*}
(p_1, \cdots, p_n)(b_1, \cdots, b_n) = (p_1 b_1, b_1^{-1}p_2 b_2, b_2^{-1}p_3 b_3,\cdots, b_{n-1}^{-1}p_n b_n)
.\end{align*}

\end{remark}

\begin{example}[?]

Take \(G = \operatorname{GL}_3({\mathbb{C}})\), so \(S = (s_1, s_2)\)
and \(w= (s_{i_1}, s_{i_2})\), then
\begin{align*}
Z_w = (P_{i_1} \times P_{i_2})/B{ {}^{ \scriptscriptstyle\times^{2} }  } = P_{i_1} \overset{\scriptscriptstyle {B} }{\times} P_{i_2}/B = P_{i_1}{ \underset{\scriptscriptstyle {B} }{\times} } P_{i_2}/B \to P_{i_1}/M \cong {\mathbb{P}}^1
,\end{align*}
so these are all bundles over \({\mathbb{P}}^1\) with fibers
\({\mathbb{P}}^1\), and are in fact Hirzebruch surfaces.

\end{example}

\begin{fact}

\envlist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(Z_w\) is an irreducible smooth variety with a
  \(P_{i_1}{\hbox{-}}\)action.
\item
  \(Z_w\to P_{i_1}/B\) is locally trivial with fiber \(Z_{w'}\) where
  \(w'\) is obtained from \(w\) by deleting the first reflection, so
  \(s' = (s_{i_2}, \cdots, s_{i_n})\).
\item
  \(Z_w \xrightarrow{\psi} Z_{w_1}\) where
  \(w_1 \coloneqq w[n-1] \coloneqq(s_1, \cdots, s_{i_n - 1})\) where
  \([p_1,\cdots, p_n] \xrightarrow{\psi} [p_1,\cdots, p_{n-1}]\). This
  admits a section
  \([p_1,\cdots, p_{n-1}] \xrightarrow{\sigma} [p_1,\cdots, p_{n-1}, 1]\).
\item
  \(Z_w\) is a projective variety.
\end{enumerate}

\end{fact}

\begin{remark}

Why projective: it's a fiber bundle of compact varieties, thus compact
and complete. A bit more goes into fully showing projectivity.

\end{remark}

\begin{definition}[?]

Define a map
\begin{align*}
m_w: Z_w &\to {\mathcal{G}}/B \\
[p_1, \cdots, p_n] &\mapsto p_1\cdots p_n B
,\end{align*}
then
\(\operatorname{im}m_w = \displaystyle\bigcup_{v\leq w} BvB/B \subseteq {\mathcal{G}}/B\).

\end{definition}

\begin{remark}

This is where the projective variety structure comes from, and we'll
discuss when the image hits Schubert varieties.

\end{remark}

\hypertarget{appendix-preliminary-notions}{%
\section{Appendix: Preliminary
Notions}\label{appendix-preliminary-notions}}

To define

\begin{itemize}
\tightlist
\item
  Sheaves

  \begin{itemize}
  \tightlist
  \item
    Coherent sheaves
  \end{itemize}
\item
  Complete variety
\item
  Homogeneous variety
\item
  Algebraic group

  \begin{itemize}
  \tightlist
  \item
    Morphisms of algebraic groups
  \item
    Reductive group
  \end{itemize}
\item
  Borel
\item
  Parabolic
\item
  Equivariant
\item
  \({\mathsf{B}}G\)

  \begin{itemize}
  \tightlist
  \item
    Some examples?
    \({\mathbb{CP}}^\infty, {\mathsf{B}}\operatorname{GL}_n({\mathbb{R}})\),
    etc.
  \end{itemize}
\item
  \({\mathsf{K}}{\hbox{-}}\)theory of an abelian category.
\item
  Segre embedding
\item
  Weyl group
\item
  Modular representation
\item
  Polar variety
\item
  Chern class
\item
  Borel-Moore homology
\item
  Relative homology
\item
  Ind-varieties and Ind-schemes
\end{itemize}

\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}