\input{"preamble.tex"}

\addbibresource{FlagVarieties.bib}

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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:} 2021-12-04 }
\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
\(H_G^* = H_G^*({\operatorname{pt}}) \coloneqq H^*_{\mathrm{sing}}({\mathbf{B}}G)\).
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}

\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\vert}~}\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}

\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 \(n=4, k=2\):

\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\vert}~}\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\vert}~}\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\vert}~}\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\vert}~}\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\vert}~}\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
  \(W = \left\{{ (w(1), w(2), \cdots, w(n) ) {~\mathrel{\Big\vert}~}w\in S_n }\right\}\).
\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\vert}~}\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,
\(\dim({\mathbb{C}}^i \cap w{\mathbb{C}}^j) = \#\left\{{ k {~\mathrel{\Big\vert}~}k\leq i, w(k) \leq j}\right\}\).
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
\(\left\{{k {~\mathrel{\Big\vert}~}k\leq j,\, w(k) \leq i}\right\}\)
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
\((1, 2)\mapsto w = (2, 4 {~\mathrel{\Big\vert}~}1, 3)\), recalling that
\((1, 2)\) has this form:

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

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

\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}

\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\vert}~}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\vert}~}\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 \mathop{\mathrm{Hom}}(U, {\mathbb{C}}) {~\mathrel{\Big\vert}~}\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\vert}~}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\vert}~}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\vert}~}\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
  \({\mathfrak{n}}^{\pm \alpha} = \bigoplus_{\alpha\in Q^+\setminus\left\{{{\operatorname{pt}}}\right\}} {\mathfrak{g}}_{\pm \alpha}\).
\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\vert}~}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\{{{\operatorname{pt}}}\right\}} {\mathfrak{r}}_{-\alpha}
}
\oplus 
\tilde{\mathfrak{r}}_0
\oplus 
\qty{
\bigoplus_{\alpha\in Q^+ \setminus\left\{{{\operatorname{pt}}}\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\vert}~}i\neq j }\right\}
&&
\left\{{f_{i, j} = (\operatorname{ad}f_i)^{1 - a_{i, j}} (f_{j}) {~\mathrel{\Big\vert}~}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
\(\Delta \coloneqq\left\{{\alpha\in Q\setminus\left\{{{\operatorname{pt}}}\right\} {~\mathrel{\Big\vert}~}{\mathfrak{g}}_\alpha \neq 0}\right\}\)
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
\(\left\{{s_i {~\mathrel{\Big\vert}~}1\leq i\leq \ell}\right\}\), 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
\(\left\{{s{~\mathrel{\Big\vert}~}s\in Y}\right\}\). 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
  \(\Delta \subseteq V\setminus\left\{{{\operatorname{pt}}}\right\}\)
  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
\(\left\{{\alpha_s {~\mathrel{\Big\vert}~}s\in S}\right\}\). 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}
\(D_{\mathbb{R}}\subseteq {\mathfrak{h}}_{\mathbb{R}} {}^{ \vee }\coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{{\mathbb{R}}}{\hbox{-}}\mathsf{Mod}}}({\mathfrak{h}}_{\mathbb{R}}, {\mathbb{R}})\)
as
\begin{align*}
D_{\mathbb{R}}\coloneqq\left\{{ \lambda\in {\mathfrak{h}}_{\mathbb{R}} {}^{ \vee }{~\mathrel{\Big\vert}~}\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\vert}~}\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
\(W_\lambda \coloneqq\left\{{w\in W {~\mathrel{\Big\vert}~}w(\lambda) = \lambda}\right\}\).
This is generated by the simple reflections it contains. Setting
\(Y = Y(\lambda) = \left\{{s_i \in S {~\mathrel{\Big\vert}~}\lambda( \alpha_i {}^{ \vee }) = 0}\right\}\),
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\vert}~}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\vert}~}\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\vert}~}{\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
  \(\left\langle{\tau_k}\right\rangle \coloneqq\left\langle{(k, k+1) {~\mathrel{\Big\vert}~}{1\leq k \leq n-1}}\right\rangle\).
\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
  \(S \subseteq \left\{{w\in W {~\mathrel{\Big\vert}~}w^2 = \operatorname{id}}\right\}\)
  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
\(T = \mathop{\mathrm{Hom}}_{\mathbb{Z}}({\mathfrak{h}}_{\mathbb{Z}}, {\mathbb{C}}^{\times})\)
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\vert}~}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\vert}~}{\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\vert}~}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{friday-september-17}{%
\section{7.1 (Friday, September 17)}\label{friday-september-17}}

\begin{remark}

See Fulton, Young Tableaux.

\end{remark}

\begin{remark}

Given \(A\) we produce \({\mathcal{G}}\) a Kac-Moody group, with
standard parabolics \(P_{\lambda} \subseteq {\mathcal{G}}\). We'll show
\(G/P_{\lambda} \hookrightarrow{\mathbb{P}}(V)\) for some projective
space over \(V\) an integrable highest weight space in
\({\mathsf{{\mathfrak{g}}}{\hbox{-}}\mathsf{Mod}}\), which is generally
an Ind-variety, and if we show it's closed it will inherit the structure
of a projective variety. Write \(V = L^{\max}( \lambda) = V_{ \lambda}\)
as a highest weight module.

Idea: for \(m_w: Z_w \to {\mathcal{G}}/B\) for \(Z_w\) a BSDH, for any
word \(w\in {\mathcal{W}}\), if \(w\in W_Y'\) is reduced, compose the
above map with \({\mathcal{G}}/B \to {\mathcal{G}}/ P_{Y}\) to get a map
\begin{align*}
m_w^Y: Z_w \to {\mathcal{G}}/P_Y
.\end{align*}

We'll show \(Z_w\) is projective, which is easier since it's an iterated
line bundle. Let \(v_0 \in V_\lambda\) (thought of in the finite type
case as a highest weight vector in the irreducible, but may generally
not coincide) consider the maps

\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5mu\iota\mkern-1.5mu}\mkern 1.5mu_V: {\mathcal{G}}&\to {\mathbb{P}}(V) \\
\iota_V: {\mathcal{G}}/P_Y &\to {\mathbb{P}}(V) \\
m_w(v_0) = \iota_{v_0} \circ m_w^Y: Z_w &\to {\mathbb{P}}(V) 
.\end{align*}

\end{remark}

\begin{theorem}[?]

\envlist

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(m_w(v_0)\) a morphism of varieties: easy to believe, hard to show!
  See the book.
\item

  \begin{align*}
  \operatorname{im}(m_w(v_0)) = \displaystyle\bigcup_{v\leq w, v\in W_Y'} BvP_Y/P_Y \subseteq {\mathcal{G}}/P_Y
  ,\end{align*}
  which is some subvariety of the flag variety which we'll define as the
  Schubert variety \(X_W^Y\).
\end{enumerate}

\end{theorem}

\begin{proposition}[5.1.3]

For \(Y \subseteq S, w\in W_Y'\), and let \(w = w_1\cdots w_k\) a
reduced decomposition, \(\ell(w) = \sum \ell(w_i)\). Let
\(Z_i \subseteq P_{w_i} \coloneqq P_{\left\{{w_i}\right\}}\) be a subset
of a simple parabolic such that \(Z_i \twoheadrightarrow P_{w_i}/B\).
\footnote{See Fulton for an explicit description, taking a Plucker
  embedding and studying actual equations.}

Then
\begin{align*}
\operatorname{im}\qty{\prod Z_i \xrightarrow{\text{mult}} G \twoheadrightarrow G/P_Y} = \displaystyle\bigcup_{v\leq w} BvP_Y/P_Y
.\end{align*}

\end{proposition}

\begin{remark}

Where does the additional condition \(v\in W_Y'\) come from in the
theorem statement? Take a Bruhat decomposition
\begin{align*}
{\mathcal{G}}/B = \displaystyle\coprod_{\substack{ v\leq w \\ v\in \dcoset{W}{W_Y}{W_Y} }} P_{Y'}vP_V
.\end{align*}

\end{remark}

\begin{example}[?]

Take \(G = \operatorname{GL}_n\), then

\begin{itemize}
\tightlist
\item
  \(\lambda\in X {}^{ \vee }(T)\)
\item
  \(\lambda(t) = t_1\cdots t_k\) for \(1\leq k\leq n\),
\item
  \(\lambda \in D_{\mathbb{Z}}\) and
  \(V_{\lambda} = \bigwedge\nolimits^k {\mathbb{C}}^n\).
\item
  \(S = \left\{{1, \cdots, \ell}\right\}\) where \(\ell = n-1\).
\item
  \(G/P_Y \subseteq {\mathbb{P}}( \bigwedge\nolimits^k {\mathbb{C}}^n)\),
\item
  \(Y( \lambda) \coloneqq\left\{{1\leq i \leq \ell {~\mathrel{\Big\vert}~}\lambda(\alpha_i {}^{ \vee }) = 0}\right\}\).
\end{itemize}

Then
\begin{align*}
\lambda &\in (1, \cdots_k, 1, 0, \cdots_{n-k}, 0)\\
\alpha_i {}^{ \vee }&= (0, \cdots, 1, -1, 0, \cdots, 0)
,\end{align*}
so we can write
\(Y( \lambda) = \left\{{1,\cdots, k-1, k+1, \cdots, n-1}\right\} = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\).
Then set
\(F^k \in {\mathbb{P}}(\bigwedge\nolimits^k {\mathbb{C}}^n) = {\operatorname{Gr}}_k({\mathbb{C}}^n)\),
so \(0 \subseteq F^k \subseteq {\mathbb{C}}^n\), and define the map
\begin{align*}
\iota_{\lambda}(F^k) = [f_1 \vee f_2 \vee\cdots \vee f_k]
,\end{align*}
where \(\left\{{f_i}\right\}\) is a choice of ordered basis.

\end{example}

\begin{fact}

Some facts about \(Z_w = \prod^B_{1\leq k\leq m} P_{i_k}/B\), recalling
the action of \(B\) given last time. Set
\(w= (s_{i_1}, \cdots, s_{i_m}) \in {\mathcal{W}}\). There is a map
\begin{align*}
\varphi: P_{i_1}\overset{\scriptscriptstyle {B} }{\times} \cdots \overset{\scriptscriptstyle {B} }{\times} P_{i_m} &\to B/B \times G/B \times \cdots \times G/B \\
[p_1,\cdots, p_m] &\mapsto [B/B, p_1 B/B, p_1p_2 B/B, \cdots, p_1\cdots p_m B/B]
.\end{align*}

Showing this is well-defined: follows from universal property of
quotients, looking at where point stabilizers are contained. Then
\begin{align*}
\operatorname{im}\varphi = 
B/B { \underset{\scriptscriptstyle {G/P_{i_1}} }{\times} }
G/B { \underset{\scriptscriptstyle {G/P_{i_2}} }{\times} }
\times \cdots 
{ \underset{\scriptscriptstyle {G/P_{i_m}} }{\times} } G/B
.\end{align*}

How to define the BSDH: construct a lattice by deleting elements in the
sequence of flags corresponding to various words, and take the
right-most flag in the result:

\begin{center}
\begin{tikzcd}
    & {i_1=n-2} &&&& {i_m = n-2} \\
    {{\mathbb{C}}^n} && {{\mathbb{C}}^{n}} && {{\mathbb{C}}^{n}} && {{\mathbb{C}}^{n}} &&& {{\mathbb{C}}^n} &&&&&&& {{\mathbb{C}}^n} \\
    {{\mathbb{C}}^{n-1}} && {A^{n-1}} && {E^{n-1}} && {F^{n-1}} &&& {{\mathbb{C}}^{n-1}} &&&& {E^{n-1}} &&& {E^{n-1}} \\
    {{\mathbb{C}}^{n-2}} && {A^{n-2}} && {E^{n-2}} && {F^{n-2}} &&& {{\mathbb{C}}^{n-2}} && {A^{n-2}} &&& {F^{n-2}} && {F^{n-2}} \\
    \vdots && \vdots && \vdots && \vdots && {} & \vdots &&& {C_2} & {D_3} &&& {D^3} \\
    {{\mathbb{C}}^1} && {A^1} && {E^1} && {F^1} &&& {{\mathbb{C}}^{1}} && {B_1} && \ddots &&& {C^2} \\
    0 && 0 && 0 && 0 &&& 0 &&&&&&& {B^1} \\
    &&&&&&&&&&&&&&&& 0
    \arrow["{\text{can differ}}", from=4-1, to=4-3]
    \arrow[Rightarrow, from=5-7, to=5-9]
    \arrow[from=3-10, to=4-12]
    \arrow[from=4-10, to=4-12]
    \arrow[from=7-10, to=6-12]
    \arrow[from=6-12, to=5-10]
    \arrow[from=6-12, to=5-13]
    \arrow[from=5-13, to=5-10]
    \arrow[from=4-12, to=5-14]
    \arrow[from=5-13, to=5-14]
    \arrow[from=4-12, to=3-14]
    \arrow[from=2-10, to=3-14]
    \arrow[from=3-14, to=4-15]
    \arrow[from=5-14, to=4-15]
    \arrow[from=5-13, to=6-14]
    \arrow["{\text{can differ}}"{description}, from=4-5, to=4-7]
    \arrow[from=8-17, to=7-17]
    \arrow[from=7-17, to=6-17]
    \arrow[from=6-17, to=5-17]
    \arrow[from=5-17, to=4-17]
    \arrow[from=4-17, to=3-17]
    \arrow[from=3-17, to=2-17]
    \arrow[from=7-3, to=6-3]
    \arrow[from=7-1, to=6-1]
    \arrow[from=4-1, to=3-1]
    \arrow[from=3-1, to=2-1]
    \arrow[from=3-3, to=2-3]
    \arrow[from=4-3, to=3-3]
    \arrow[from=3-5, to=2-5]
    \arrow[from=4-5, to=3-5]
    \arrow[from=4-7, to=3-7]
    \arrow[from=3-7, to=2-7]
    \arrow[from=3-10, to=2-10]
    \arrow[from=4-10, to=3-10]
    \arrow[from=7-10, to=6-10]
    \arrow[from=7-7, to=6-7]
    \arrow[from=7-5, to=6-5]
\end{tikzcd}
\end{center}

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

Here the word is \((i_{n-2}, i_1, i_2, i_{3}, i_{n-1}, i_{n-2})\).

\end{fact}

\hypertarget{equivariant-mathsfkhbox-theory-wednesday-september-22}{%
\section{\texorpdfstring{Equivariant \({\mathsf{K}}{\hbox{-}}\)theory
(Wednesday, September
22)}{Equivariant \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory (Wednesday, September 22)}}\label{equivariant-mathsfkhbox-theory-wednesday-september-22}}

\begin{remark}

The setup: \(G\curvearrowright X\) a topological group acting on a
space.

\begin{itemize}
\item
  Gelfand (30s): replace \(X\) with a topological vector space \(T\),
  e.g.~``generalized functions'' on \(X\). This linearises the problem,
  but is usually something like a infinite dimensional Hilbert space.
\item
  Harish-Chandra, Vogan: Replace \(T\) with an algebraic object (usually
  finite-dimensional) and apply \({\mathsf{K}}{\hbox{-}}\)theory. Here
  \({\mathsf{K}}{\hbox{-}}\)theory simplifies the problem, since all
  invariants that are additive on exact sequences can be recovered from
  it.
\end{itemize}

Classical literature on this is phrased in terms of \(X\) a
\emph{separated algebraic space}, since even nice quotients of varieties
are often not again varieties. We'll assume \(X\) is an algebraic
variety, automatically separated, and quasiprojective. This will imply
that \(X \subseteq G/P\) embeds into a flag variety, e.g.~for
\(G = \operatorname{GL}_n\) and \(P\) a parabolic this covers
\({\mathbb{P}}^n\). For us, projective will mean that
\(X \subseteq G/P\) is closed, which will turn out to admit ample line
bundles.

\end{remark}

\begin{definition}[?]

Let \(({-})^ {\operatorname{gp} }\) denote taking the Grothendieck
group, then
\begin{align*}
G_0(G, X) \coloneqq{\mathsf{Coh}}^G(X)^ {\operatorname{gp} } \\
{\mathsf{K}}_0(G, X) \coloneqq{ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }^G_{/ {X}} 
,\end{align*}
i.e.~the \(G{\hbox{-}}\)equivariant coherent sheaves and vector bundles
respectively.

\end{definition}

\begin{remark}

Note that vector bundles don't form an abelian category -- here instead
you take the additive monoid generated by addition of vector bundle.
However coherent sheaves do form an abelian category, so this denotes
the usual Grothendieck group for abelian categories. Of modern interest:
split Grothendieck groups, triangulated, etc.

Here one should think of \({\mathsf{G}}\) as something analogous to
Borel-Moore homology, and \({\mathsf{K}}\) is closer to cohomology. Note
that in classical settings, one could cap against the fundamental class
to get a map between them.

\end{remark}

\begin{proposition}[?]

If \(X\) is a smooth \(G{\hbox{-}}\)variety admitting an equivariant
ample line bundle, then there is an isomorphism
\begin{align*}
{\mathsf{K}}_0(G, X) \xrightarrow{?} G_0(G, X)
.\end{align*}

\end{proposition}

\begin{remark}

This map records how a vector bundle can be regarded as a coherent
sheaf! For the rest of today, we'll assume \(X\) admits a
\(G{\hbox{-}}\)equivariant ample line bundle and refer to this as
condition \(\star\). If this proposition holds, notationally we'll
always write \({\mathsf{K}}^G(X) = {\mathsf{K}}_0(X) = G_0(X)\).

\end{remark}

\begin{example}[?]

Consider the coherent sheaf
\({\mathcal{O}}_x{ {}^{ \scriptscriptstyle\oplus^{n} } }\), which should
correspond to the trivial bundle \(X\times {\mathbb{C}}^n \to X\). If
\(\xi\) is a vector bundle, then the sheaf of sections is a locally free
coherent sheaf.

\end{example}

\begin{proposition}[?]

Every \(G{\hbox{-}}\)equivariant coherent sheaf
\({\mathcal{F}}\in {\mathsf{Coh}}^G(X)\) on \(X\) admits a finite
resolution by \(G{\hbox{-}}\)equivariant locally free sheaves of finite
type.

\end{proposition}

\begin{example}[?]

\envlist

\begin{itemize}
\item
  If \(G =1\), then admitting an ample line bundle as above is
  equivalent to \(X \subseteq G/P\) being a subvariety. Then
  \({\mathsf{K}}^G(X) = {\mathsf{K}}(X)\), the algebraic
  \({\mathsf{K}}{\hbox{-}}\)theory, of \(X\).
\item
  If \(X = {\operatorname{pt}}\), it is smooth, and
  \({\mathsf{K}}^G(X) = R(G)\), the representation ring of \(G\). This
  holds for \(G\) any linear algebraic group.
\end{itemize}

\end{example}

\begin{remark}

So this mixes usual \({\mathsf{K}}{\hbox{-}}\)theory and representation
theory! It turns out that for \(X={\operatorname{pt}}\), there is an
equivalence of categories
\({\mathsf{Coh}}^G({\operatorname{pt}}) = { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }^G = {\mathsf{G}{\hbox{-}}\mathsf{Mod}}^{\mathrm{fd}}\).

If \(X\) is projective and \(G\) is semisimple, then \(\star\) is true.
If \(E\to X\) is a \(G{\hbox{-}}\)vector bundle on \(X\) smooth
projective, then we'll write \({\mathsf{K}}^G(X)\) for
\({\mathsf{G}}_0(G, X) = {\mathsf{K}}_0(G, X)\).

\end{remark}

\begin{lemma}[?]

Every \({\mathcal{F}}\in {\mathsf{Coh}}^G(X)\) is a quotient of a
\(G{\hbox{-}}\)equivariant locally free sheaf \({\mathcal{E}}\) of
finite type on \(X\).

\end{lemma}

\begin{quote}
See proof: Borho, Byrlinksi, MacPherson. Geometric perspective on ring
theory?
\end{quote}

\begin{remark}

Let \(G \in {\mathsf{Alg}}{\mathsf{Grp}}\) be linear acting on
\(V\in {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) possibly infinite
dimensional. This is common, e.g.~when \(G\) consists of regular
functions. This is infinite dimensional, but not so bad -- it's not
quite as big as a Hilbert space. We'll say the action is
\textbf{algebraic} if it acts locally finitely: the \(G{\hbox{-}}\)orbit
of any vector should be a finite dimensional subspace. Consider
\(\mathop{\mathrm{Maps}}(G, M) \coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{Set}}}(G, M) = \left\{{ f:G\to M}\right\}\)
with no conditions at all on the functions. There is a subspace of
``regular functions with coefficients in \(M\)'', using the following
well-defined map:
\begin{align*}
{\mathbb{C}}[G] \otimes_{\mathbb{C}}M &\to \mathop{\mathrm{Maps}}(G, M) \\
\sum f_i \otimes m_i &\mapsto \sum f(g_i) m_i
,\end{align*}
using that the \(f(g_i)\) are scalars in \(M\).

For a fixed \(m\), there is a \(G{\hbox{-}}\)action \(g\mapsto gm\), and
so letting \(m\) vary yields a map
\(M \xrightarrow{a_M} \mathop{\mathrm{Maps}}(G, M)\).

\begin{claim}

\(G\) acts algebraically on \(M\) iff
\(\operatorname{im}a_m \subseteq {\mathbb{C}}[G]\otimes_{\mathbb{C}}M\).

\end{claim}

If the action is algebraic, take \(G_m \subseteq V \subseteq M\) with
\(V\) a \(G{\hbox{-}}\)stable finite dimensional subspace. Expanding in
a basis and writing \(g\mapsto gm\) in this basis yields the \(f_i\),
which are regular.

\end{remark}

\begin{lemma}[?]

If \({\mathcal{F}}\in {\mathsf{Coh}}^G(X)\), then
\({{\Gamma}\qty{X; {\mathcal{F}}} }\) has the natural structure of an
algebraic \(G{\hbox{-}}\)module.

\end{lemma}

\hypertarget{localization-in-equivariant-mathsfkhbox-theory-friday-september-24}{%
\section{\texorpdfstring{Localization in Equivariant
\({\mathsf{K}}{\hbox{-}}\)theory (Friday, September
24)}{Localization in Equivariant \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory (Friday, September 24)}}\label{localization-in-equivariant-mathsfkhbox-theory-friday-september-24}}

\hypertarget{localization-theorems}{%
\subsection{Localization Theorems}\label{localization-theorems}}

\begin{quote}
Reference: Thomason.
\end{quote}

\begin{definition}[Localization theorems]

Suppose \(A\in {\mathsf{Ab}}{\mathsf{Alg}}{\mathsf{Grp}}\) is reductive,
and \(X \subseteq G/P\) is contained in a flag variety (so \(X\) is
quasiprojective). Fix \(a\in A\), and consider the fixed point set
\(X^a\) and the inclusion \(\iota: X^a \xrightarrow{\subseteq } X\).
We'll say the \textbf{localization theorem holds for \(X\)} if the
following induced hom is an isomorphism:
\begin{align*}
i_*: {\mathsf{K}}^A(X^a) \left[ { \scriptstyle { {{\mathfrak{m}}_a}^{-1}}  } \right] \to {\mathsf{K}}^A(X) \left[ { \scriptstyle { {{\mathfrak{m}}_a}^{-1}}  } \right]
.\end{align*}

\end{definition}

\begin{remark}

Thomason shows that this is true in this situation. Recall that we
identified \(R(A) = {\mathsf{K}}^A({\operatorname{pt}})\). Taking the
trace of a representation yields a map
\(R(A) \hookrightarrow{\mathbb{C}}[A]\), the ring of regular functions.
For varieties, we can obtain \({\mathcal{O}}_{X, x}\) by localizing
rings at their maximal ideals, thinking of these as functions on \(X\).
Let
\begin{align*}
R_a &\coloneqq R(A) \left[ { \scriptstyle { { \qty{ R(A)\setminus{\mathfrak{m}}_a} }^{-1}}  } \right] \\
M_a &\coloneqq R(A) \otimes_{R(A)} M
.\end{align*}

\end{remark}

\hypertarget{proper-pushforward}{%
\subsection{Proper Pushforward}\label{proper-pushforward}}

\begin{remark}

We'll need proper maps for the ever-popular \emph{decomposition
theorem}. However, almost every scheme we use in this class will be
reduced, although one does rarely have to worry about this.

\end{remark}

\begin{definition}[Proper Maps (and prerequisite notions)]

\textbf{Pullbacks} are universal with respect to the following squares,
and have a concrete description for us:

\begin{center}
\begin{tikzcd}
    {\left\{{(x, z) \in X\times Z {~\mathrel{\Big\vert}~}f(x) = z(g)}\right\}} \\
    {X{ \underset{\scriptscriptstyle {Y} }{\times} }Z} && Z \\
    \\
    X && Y
    \arrow["g", from=2-3, to=4-3]
    \arrow["f"', from=4-1, to=4-3]
    \arrow["{g'}"', from=2-1, to=4-1]
    \arrow["{f'}", from=2-1, to=2-3]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-1, to=4-3]
    \arrow[Rightarrow, no head, from=1-1, to=2-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwxLCJYXFxmaWJlcnByb2R7WX1aIl0sWzIsMSwiWiJdLFswLDMsIlgiXSxbMiwzLCJZIl0sWzAsMCwiXFx0c3soeCwgeikgXFxpbiBYXFx0aW1lcyBaIFxcc3QgZih4KSA9IHooZyl9Il0sWzEsMywiZyJdLFsyLDMsImYiLDJdLFswLDIsImcnIiwyXSxbMCwxLCJmJyJdLFswLDMsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFs0LDAsIiIsMCx7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV1d}{Link
to Diagram}
\end{quote}

The \textbf{diagonal} is the unique morphism
\(\Delta: X\to X{ \underset{\scriptscriptstyle {Y} }{\times} } X\) whose
compositions with projections are the identity:

\begin{center}
\begin{tikzcd}
    X \\
    \\
    && {X{ \underset{\scriptscriptstyle {Y} }{\times} } X} && X \\
    \\
    && X && Y
    \arrow[from=3-5, to=5-5]
    \arrow[from=5-3, to=5-5]
    \arrow[from=3-3, to=5-3]
    \arrow[from=3-3, to=3-5]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-3, to=5-5]
    \arrow["{\operatorname{id}_X}", from=1-1, to=3-5]
    \arrow["{\operatorname{id}_X}"', from=1-1, to=5-3]
    \arrow["\Delta"{description}, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMiwyLCJYXFxmaWJlcnByb2R7WX0gWCJdLFs0LDIsIlgiXSxbMiw0LCJYIl0sWzQsNCwiWSJdLFswLDAsIlgiXSxbMSwzXSxbMiwzXSxbMCwyXSxbMCwxXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbNCwxLCJcXGlkX1giXSxbNCwyLCJcXGlkX1giLDJdLFs0LDAsIlxcRGVsdGEiLDFdXQ==}{Link
to Diagram}
\end{quote}

A morphism is \textbf{separated} if the diagonal is a closed embedding.

A morphism \(f:X\to Y\) is \textbf{universally closed} if for any
\(g:Z\to Y\), the base change
\(f': X{ \underset{\scriptscriptstyle {Y} }{\times} } Z\to Z\) is a
closed morphism. This replaces the notion of ``\(K\) compact
\(\implies f^{-1}(K)\) compact'' for analytic varieties.

A morphism \(f\) is \textbf{proper} if \(f\) is separated, finite type,
and universally closed.

\end{definition}

\begin{example}[?]

\envlist

\begin{itemize}
\tightlist
\item
  Closed embeddings are proper, and open maps are usually not.
\item
  If \(f\) is proper, its base change \(f'\) is always proper.
\item
  Compositions of proper morphisms are again proper.
\item
  Any morphism between projective varieties is proper.
\end{itemize}

\end{example}

\begin{theorem}[18.8.1, Rising Sea]

Let \(f:X\to Y\) be proper and \({\mathcal{F}}\in {\mathsf{Coh}}(X)\).
Note that \({{\Gamma}\qty{X, {-}} }\) is exact and \({\mathsf{Coh}}(X)\)
is abelian, so we can take its derived functor. Let
\(f_*: {\mathsf{Sh}}_{/ {X}} \to {\mathsf{Sh}}_{/ {Y}}\), then e.g.~
\begin{align*}
{\mathbb{R}}^i f_* {\mathcal{F}}(U) = H^i(f^{-1}(U); {\mathcal{F}})
.\end{align*}

This satisfies several properties:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \({\mathbb{R}}^if_*: {\mathsf{Coh}}(X) \to {\mathsf{Coh}}(Y)\) is a
  covariant functor. Without properness, one can just replace
  \({\mathsf{Coh}}\) with \({\mathsf{QCoh}}\).
\item
  \({\mathbb{R}}^0 f_* = f_*\)
\item
  A SES \(0\to{\mathcal{F}}_1\to{\mathcal{F}}_2\to{\mathcal{F}}_3\to 0\)
  induces a LES.
\end{enumerate}

\end{theorem}

\begin{theorem}[Rising Sea, 18.8.5]

If \(f:X\to Y\) is a proper projective morphism, then
\({\mathbb{R}}^{i>d} f_* {\mathcal{F}}= 0\) for \(d\) defined as the
maximum dimension of the fiber,
\(d\coloneqq\max_{y\in Y} \dim f^{-1}(y)\).

\end{theorem}

\begin{definition}[Proper Pushforward]

Let \(X, Y\) be arbitrary quasiprojective varieties and \(f:X\to Y\) be
proper and \(G{\hbox{-}}\)equivariant. Then there is a natural direct
image morphism \(f_*: {\mathsf{K}}^G(X) \to {\mathsf{K}}_G(Y)\).

We define it as follows: note that a map such as
\(f_*([{\mathcal{F}}]) \coloneqq[f_* {\mathcal{F}}]\) won't necessarily
be well-defined, since SESs are additive in the Grothendieck group. For
\({\mathcal{F}}\in {\mathsf{Coh}}^G(X)\), then it turns out that
\({\mathbb{R}}f_* {\mathcal{F}}\in {\mathsf{Coh}}^G(Y)\) and the higher
direct images vanish in large enough degree. We then define
\begin{align*}
f_*: {\mathsf{K}}^G(X) &\to {\mathsf{K}}_G(Y) \\
[{\mathcal{F}}] &\mapsto \sum (-1)^i [{\mathbb{R}}^i f_* {\mathcal{F}}]
.\end{align*}

\end{definition}

\begin{example}[?]

Let \(G\) be connected reductive with \(A \coloneqq T\) a maximal torus,
which is abelian reductive. Then take \(a\in A\) a \emph{regular}
element, so \(X^a = X^T\). In our case, \(X^T = W_Y'\), and
\(X = G/P_Y\). Then \({\mathsf{K}}{\hbox{-}}\)theory is concentrated on
the fixed locus:
\begin{align*}
i_* {\mathsf{K}}^T(X^T) \left[ { \scriptstyle { {{\mathfrak{m}}_a}^{-1}}  } \right] { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}^T(X) \left[ { \scriptstyle { {{\mathfrak{m}}_a}^{-1}}  } \right]
.\end{align*}

\end{example}

\hypertarget{line-bundles-on-mathcalxy-monday-september-27}{%
\section{\texorpdfstring{Line Bundles on \({\mathcal{X}}^Y\) (Monday,
September
27)}{Line Bundles on \{\textbackslash mathcal\{X\}\}\^{}Y (Monday, September 27)}}\label{line-bundles-on-mathcalxy-monday-september-27}}

\begin{remark}

Notation: \({\mathcal{X}}\) will denote a Kac-Moody flag variety, and
\(X\) a usual flag variety. For any \(\lambda \in D_Y^0\), define the
algebraic line bundle \({\mathcal{L}}(-\lambda) \to {\mathcal{X}}^Y\) to
be the pullback of the tautological bundle on
\({\mathbb{P}}(L^{\max}(\lambda))\) via the morphism
\(\iota_\lambda: {\mathcal{X}}^Y \to {\mathbb{P}}(L^{\max} (\lambda))\).
Recall that we defined \(Y{\hbox{-}}\)regular weights to get an
embedding into a flag variety.

Let \(X\) be a finite dimensional variety, then a vector bundle on \(X\)
is a map \({\mathcal{E}}\xrightarrow{\pi} X\) with each fiber a
\({\mathbb{C}}{\hbox{-}}\)module and for all \(x\in X\) there exists an
open \(U \subseteq X\) and a homeomorphism
\(\phi: U \times {\mathbb{C}}^n \to \pi^{-1}(U)\) over \(U\), so the
following diagram commutes:

\begin{center}
\begin{tikzcd}
    {\pi^{-1}(U)} && {U\times {\mathbb{C}}^n} \\
    \\
    & U
    \arrow["\pi"', from=1-1, to=3-2]
    \arrow["{{\operatorname{pr}}_1}", from=1-3, to=3-2]
    \arrow["\phi", from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHBpXFxpbnYoVSkiXSxbMiwwLCJVXFx0aW1lcyBcXENDXm4iXSxbMSwyLCJVIl0sWzAsMiwiXFxwaSIsMl0sWzEsMiwiXFxwcl8xIl0sWzAsMSwiXFxwaGkiXV0=}{Link
to Diagram}
\end{quote}

We refer to \(\phi\) as a trivialization. Writing
\(U_{12} \coloneqq U_1 \cap U_2\), given trivializations over \(U_i\) we
require that the trivializations on \(U_{12}\) are related by an element
\(T_{12} \in \operatorname{GL}_n\), and the induced map
\(U_{12} \times {\mathbb{C}}^n{\circlearrowleft}\) are essentially given
by matrices with entries given by functions on \(U_{12}\) The key is
that these satisfy the cocycle condition:
\begin{align*}
T_{kj}\mathrel{\Big|}_{U_{ijjk}} T_{ji} \mathrel{\Big|}_{U_{ijk}} = T_{ki}\mathrel{\Big|}_{U_{ijk}}
.\end{align*}

Given a vector bundle, set \({\mathcal{F}}\) to be the sheaf of sections
of \(\pi: {\mathcal{E}}\to X\). If for example \(U \subseteq X\) is
trivializable, then \({{\Gamma}\qty{U, {\mathcal{F}}} }\) are
\(n{\hbox{-}}\)tuples of functions \(U\to {\mathbb{C}}\), so
\({ \left.{{{\mathcal{F}}}} \right|_{{U}} }\cong {\mathcal{O}}_U{ {}^{ \scriptscriptstyle\oplus^{n} } }\),
making it locally free.

\end{remark}

\begin{proposition}[about locally free sheaves]

Given a vector bundle, set \({\mathcal{F}}\) to be the sheaf of sections
of \(\pi: {\mathcal{E}}\to X\). Then

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  If \({\mathcal{F}}\) is locally free, then
  \(\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}_{/ {X}} }({\mathcal{F}}, {\mathcal{O}}_X) \in {\mathsf{Sh}}_{/ {X}}\)
  is locally free.
\item
  If \(n=1\), then
  \({\mathcal{F}}\otimes{\mathcal{F}} {}^{ \vee }\cong {\mathcal{O}}_X\),
  making it an invertible sheaf under the monoidal tensor product.
\item
  Pullbacks of locally free sheaves are again locally free:
\end{enumerate}

\begin{center}
\begin{tikzcd}
Z{ \underset{\scriptscriptstyle {X} }{\times} } {\mathcal{E}}
  \ar[r] 
  \ar[d]
& 
{\mathcal{E}}
  \ar[d] 
\\
Z
  \ar[r]
& 
X 
\end{tikzcd}
\end{center}

where we equivalently write \(f^* {\mathcal{F}}\).

\end{proposition}

\begin{remark}

How to think about a flag variety: given \(w\in W_Y'\) and
\(U_W \subseteq X^Y\), so \(U^- \subseteq G/P\). Then
\(\left\{{U_w}\right\}_{w\in W_Y'} \rightrightarrows X^Y\) is an open
cover with \(U_w \cong {\mathbb{C}}^{\ell(w_0')}\) with \(w_0\) the
longest element and \(w_0'\) is a minimal coset representative. If
\(v\in U_w \iff v=w\) for any \(T{\hbox{-}}\)fixed point \(v\), so
there's only one such fixed point in every open. We have elements
\(wP/p \in X^Y\), so

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-09-27_14-18.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{remark}

\begin{example}[?]

For \(G \coloneqq{\operatorname{SL}}_{n+1}\), we have
\(Y = \left\{{2,\cdots, n}\right\}, W = S_{n+1} = \left\{{(w_0 \cdots w_n)}\right\}\)
and the minimal length representatives have increasing coordinates, so
we get
\begin{align*}
W_Y' = \left\{{(0 \mathrel{\Big|}1,2\cdots, n), (1 \mathrel{\Big|}0, 2, \cdots, n), \cdots , (n\mathrel{\Big|}0,1\cdots, n-1)}\right\}
.\end{align*}
For every \(i\in W_Y' = \left\{{0, \cdots, n}\right\}\), we have
\(U_i \subseteq X^Y \subseteq G/P_Y\). We can obtain
\({\mathbb{P}}^n \cong \leftquotient{{\mathbb{C}}^{\times}}{{\mathbb{C}}^{n+1}}\),
which is \(G/P^Y = X^Y\) here. So we can take
\(U_i \coloneqq\left\{{ {\left[ {x_0,\cdots, x_n} \right]} {~\mathrel{\Big\vert}~}x_i\neq 0 }\right\} \subseteq {\mathbb{C}}^n\),
which is dimension \(n\) since the longest element is
\((n \mathrel{\Big|}0,1,\cdots, n-1)\).

\end{example}

\begin{example}[?]

Let \(k\in {\mathbb{Z}}\), we'll define
\({\mathcal{O}}_{{\mathbb{P}}^n}(k)\), a line bundle on
\({\mathbb{P}}^n\). Taking \(n=1\) to get \({\operatorname{SL}}_2\) and
\({\mathbb{P}}^1\) above, we have \(W_Y' = \left\{{0, 1}\right\}\) and
\({\mathbb{C}}= U_1 = \operatorname{Spec}{\mathbb{C}}[x_{0/1}]\) and
\(U_0 = \operatorname{Spec}{\mathbb{C}}[x_{0/1}]\), then on their
intersection we have \(x_{0/1} = x_{1/0}^{-1}\). So transitioning
\(U_0\to U_1\) is given by \(x_{0/1}^k = x_{1/0}^{-k}\), and
\(U_1\to U_0\) by \(x_{1/0}^k = x_{0/1}^{-k}\), which defines a line
bundle denoted \({\mathcal{E}}\coloneqq{\mathcal{O}}(k)\). What are the
global sections \({{\Gamma}\qty{{\mathbb{P}}^1; {\mathcal{O}}(k)} }\)?
This requires \(f(x_{0/1}^{-1}) x_{0/1}^k = g(x_{0/1})\), so the global
sections are \({\mathbb{C}}[x,y]_{k}\) the homogeneous polynomials of
degree \(k\). One can check that
\(\dim {{\Gamma}\qty{{\mathbb{P}}^n; {\mathcal{O}}(k)} } = {n+k \choose k}\).

\end{example}

\begin{remark}

Next time: we'll try to match these up with line bundles of the form
\(G \overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_\lambda\).

\end{remark}

\hypertarget{wednesday-september-29}{%
\section{Wednesday, September 29}\label{wednesday-september-29}}

\begin{remark}

Ch. 7 and 8 in Kumar: algebraic vector bundles, particularly line
bundles on ind-varieties. Let \({\mathcal{E}}\xrightarrow{\pi} X\) be an
algebraic vector bundle, so there are local trivializations:

\begin{center}
\begin{tikzcd}
\pi^{-1}(U) 
  \ar[rd, "\pi"]
  \ar[rr, "{\operatorname{pr}}_1"] 
& 
& 
U\times {\mathbb{C}}^n
  \ar[ld, ""] 
\\
& 
U 
& 
\end{tikzcd}
\end{center}

i.e.~these look like projections onto the first coordinate of an actual
product on sufficiently small sets. We write
\({\mathcal{E}}_x \coloneqq\pi^{-1}(x)\). The key data: transition
functions.

Our first examples were \({\mathcal{O}}_{{\mathbb{P}}^n}(k)\),
particularly for \(n=1\).

\end{remark}

\begin{remark}

Equivariant coherent sheaves yields algebraic representations by taking
global sections. Kumar uses character formulas to compute global
sections.

\end{remark}

\begin{definition}[Equivariant vector bundles]

For \(G\in {\mathsf{Alg}}{\mathsf{Grp}}\) is linear (and e.g.~connected
reductive), if \(\pi\) is \(G{\hbox{-}}\)equivariant and \(G\) maps
\({\mathcal{E}}_x\to {\mathcal{E}}_{gx}\) \emph{linearly}, then \(\pi\)
yields an \textbf{equivariant vector bundle}.

\end{definition}

\begin{remark}

For \(G\) connected reductive and \(T \subseteq G\) a maximal torus, a
character \(\lambda \in X^*(T)\) is a map
\(\lambda: T\to {\mathbb{C}}^{\times}\), and using
\(T \subseteq B \subseteq G\) we get a representation
\(\lambda: B\to {\mathbb{C}}^{\times}\) of the Borel. We then define
\begin{align*}
G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda} \coloneqq(G\times {\mathbb{C}})/B
.\end{align*}
There is a map

\begin{center}
\begin{tikzcd}
    {G\times {\mathbb{C}}} && {G\overset{\scriptscriptstyle {B} }{\times}{\mathbb{C}}_{\lambda}} && {[g, z]} \\
    \\
    G && {G/B} && {gB/B}
    \arrow["{{\operatorname{pr}}_1}", from=1-1, to=3-1]
    \arrow["{{-}/B}", from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[dashed, from=1-1, to=1-3]
    \arrow[maps to, from=1-5, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMiwwLCJHXFxtaXh7Qn1cXENDX3tcXGxhbWJkYX0iXSxbMiwyLCJHL0IiXSxbNCwwLCJbZywgel0iXSxbNCwyLCJnQi9CIl0sWzAsMCwiR1xcdGltZXMgXFxDQyJdLFswLDIsIkciXSxbNCw1LCJcXHByXzEiXSxbNSwxLCJcXHdhaXQvQiJdLFswLDFdLFs0LDAsIiIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsyLDMsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=}{Link
to Diagram}
\end{quote}

Even better, if
\(Y = \left\{{1\leq i\leq \ell {~\mathrel{\Big\vert}~}{\left\langle {\lambda},~{\alpha_i {}^{ \vee }} \right\rangle} }\right\}\)
then taking \(\lambda \in D_Y^0\) so
\(\lambda: P \to {\mathbb{C}}^{\times}\) yields a map
\(G\overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_{ \lambda} \xrightarrow{\pi} G/P\)
where \(G/P \supseteq U_w\). Write \(P = LU\) and \(P^- = LU^-\) for
\(L\) the Levi and \(U^\pm\) the unipotent radical and its opposite:

\includegraphics{figures/2021-09-29_14-08-16.png}

There is an embedding
\begin{align*}
U^- &\hookrightarrow G/P\\
u &\mapsto uP/P
.\end{align*}
For \(w\in W_Y'\), we have
\begin{align*}
\eta_w: {}^w U^- &\to G/P \\
wuw^{-1}&\mapsto wuP/P
,\end{align*}
and \({}^w U^- = wU^- w^{-1}\) for \(w\in W= N_G(T)/T\).

\end{remark}

\begin{example}[?]

Let \({\mathbb{P}}^1 = G/P\) for \(G= {\operatorname{SL}}_2\). Here
\(W = \left\{{e, s}\right\}\cong C_2\) and
\(S = \left\{{s}\right\} \supseteq Y\), and we want \(Y = \emptyset\).
Any \(\lambda \in X^*(T)\) needs to be orthogonal to
\(\alpha {}^{ \vee }\). We can take a realization
\({\operatorname{SL}}_2({\mathbb{C}}, \left\{{2}\right\}, \left\{{1}\right\})\)
which yields \(X^*(T) = {\mathbb{Z}}\). So
\({\left\langle { \lambda},~{ \alpha {}^{ \vee }} \right\rangle} = 0 \iff 1\cdot \lambda \neq 0\),
forcing \(\lambda\neq 0\) for this to be a flag variety. For
\(\lambda = k\), we have
\(\lambda \cdot { \begin{bmatrix}  {t} & {0} \\  {0} & {t^{-1}} \end{bmatrix} } = t^k\).
We get a line bundle
\(G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda} \xrightarrow{\pi} G/B={\mathbb{P}}^1\),
how does this compare to \({\mathcal{O}}_{{\mathbb{P}}^1}(k)\)? The flag
varieties look like the following:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-09-29_14-17.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Here \(s, e\) are the two \(T{\hbox{-}}\)fixed points. We have
\(U_s \cap U_e \cong {\mathbb{C}}^{\times}\), and we'll replace
\(U_s \to {}^s U^-\) and \(U_e \to {}^{e}U^- = U^-\). The transition
functions read:

\begin{center}
\begin{tikzcd}
    {{}^{s}U^- \times {\mathbb{C}}} && {\pi^{-1}(U_s)} && {\pi^{-1}(U_e)} && {U^-\times {\mathbb{C}}} \\
    \\
    & {U_s} &&&& {U_e}
    \arrow["\pi", from=1-3, to=3-2]
    \arrow["{{\operatorname{pr}}_1}"', from=1-1, to=3-2]
    \arrow["\cong", from=1-1, to=1-3]
    \arrow["\cong", from=1-5, to=1-7]
    \arrow["{{\operatorname{pr}}_1}", from=1-7, to=3-6]
    \arrow["\pi"', from=1-5, to=3-6]
    \arrow["{\text{on }U_e \cap U_s}"', from=1-3, to=1-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMiwwLCJcXHBpXFxpbnYoVV9zKSJdLFswLDAsInt9XntzfVVeLSBcXHRpbWVzIFxcQ0MiXSxbNCwwLCJcXHBpXFxpbnYoVV9lKSJdLFs2LDAsIlVeLVxcdGltZXMgXFxDQyJdLFsxLDIsIlVfcyJdLFs1LDIsIlVfZSJdLFswLDQsIlxccGkiXSxbMSw0LCJcXHByXzEiLDJdLFsxLDAsIlxcY29uZyJdLFsyLDMsIlxcY29uZyJdLFszLDUsIlxccHJfMSJdLFsyLDUsIlxccGkiLDJdLFswLDIsIlxcdGV4dHtvbiB9VV9lIFxcaW50ZXJzZWN0IFVfcyIsMl1d}{Link
to Diagram}
\end{quote}

We have \(U_s \cap U_e \cong {\mathbb{C}}^{\times}\), so what map
\({\mathbb{C}}^{\times}{\circlearrowleft}\) do we get? Consider
\(U^-B/B \cap sU^-B/B\), so
\begin{align*}
u_{\alpha} (x) = {
\begin{bmatrix}
  {1} & {x} 
\\
  {0} & {1}
\end{bmatrix}
} && u_{-\alpha}(x) = {
\begin{bmatrix}
  {1} & {0} 
\\
  {x} & {1}
\end{bmatrix}
}
.\end{align*}
Then and \(u_{-\alpha}(x) = s u_\alpha(-x)s^{-1}\), so
\begin{align*}
u_{-\alpha}(x) B &= u_{- \alpha}(y) B \\
s u_{\alpha}(-x)s^{-1}B &= s u_{- \alpha}(y) B \\
u_{ \alpha}(-x) s^{-1}B &= u_{-\alpha} B
.\end{align*}

Now check that
\begin{align*}
{
\begin{bmatrix}
  {1} & {-x} 
\\
  {0} & {1}
\end{bmatrix}
} {
\begin{bmatrix}
  {0} & {-1} 
\\
  {1} & {0}
\end{bmatrix}
} = {
\begin{bmatrix}
  {1} & {0} 
\\
  {y} & {1}
\end{bmatrix}
} {
\begin{bmatrix}
  {a} & {b} 
\\
  {0} & {a^{-1}}
\end{bmatrix}
}&& \text{for some }{
\begin{bmatrix}
  {a} & {b} 
\\
  {0} & {a^{-1}}
\end{bmatrix}
}\in B \\ \\
{
\begin{bmatrix}
  {-x} & {-1} 
\\
  {1} & {0}
\end{bmatrix}
} = {
\begin{bmatrix}
  {a} & {b} 
\\
  {ay} & {yb+a^{-1}}
\end{bmatrix}
}
,\end{align*}
so we have \(-x=y^{-1}\). Thus
\begin{align*}
T_{es}U_s^{\times}\times {\mathbb{C}}&\to U_e^{\times}\times {\mathbb{C}}\\
(x, z) &\mapsto (x^{-1}, x^{-k} z) \\ \\
T_{se}U_e^{\times}\times {\mathbb{C}}&\to U_s^{\times}\times {\mathbb{C}}\\
(x, z) &\mapsto (x^{-1}, x^{-k} z)
.\end{align*}

\begin{quote}
These computations are hard, even in the case of
\({\operatorname{SL}}_2\)! Perhaps a motivation for having character
formulas.
\end{quote}

We then identify
\(G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda} \xrightarrow{\pi} G/B\)
with \({\mathcal{O}}(-k)\), and
\({\mathcal{L}}( \lambda) = G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}\).

\end{example}

\hypertarget{kumar-ch.-8-demazure-character-formulas-friday-october-01}{%
\section{Kumar Ch. 8: Demazure Character Formulas (Friday, October
01)}\label{kumar-ch.-8-demazure-character-formulas-friday-october-01}}

\begin{remark}

For any \(\lambda \in D_Y^0\) define the algebraic line bundle
\({\mathcal{L}}^Y(- \lambda)\) over \(X^Y = {\mathcal{G}}/P_Y\) to be
the following pullback:

\begin{center}
\begin{tikzcd}
    {{\mathcal{G}}/P_y} && \eta \\
    \\
    {{\mathcal{X}}^Y} && {{\mathbb{P}}(L^{\max} ( \lambda ))}
    \arrow["{\iota_{\lambda}}", from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
    \arrow[from=1-1, to=3-1]
    \arrow[from=1-1, to=1-3]
    \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jZy9QX3kiXSxbMiwwLCJcXGV0YSJdLFsyLDIsIlxcUFAoTF57XFxtYXh9ICggXFxsYW1iZGEgKSkiXSxbMCwyLCJcXG1jeF5ZIl0sWzMsMiwiXFxpb3RhX3tcXGxhbWJkYX0iXSxbMSwyXSxbMCwzXSxbMCwxXSxbMCwyLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=}{Link
to Diagram}
\end{quote}

Let \(H^Y = {\operatorname{Gr}}_k({\mathbb{C}}^n) = G/P^Y\) for
\(G\coloneqq\operatorname{GL}_n\).

\end{remark}

\begin{definition}[The Tautological Bundle]

Then define a vector bundle
\begin{align*}
{\mathcal{E}}\coloneqq\left\{{ (x, v) \in X^Y \times {\mathbb{C}}^n {~\mathrel{\Big\vert}~}v\in x}\right\} = \left\{{(E, v) \in {\operatorname{Gr}}_k({\mathbb{C}}^n) \times {\mathbb{C}}^n {~\mathrel{\Big\vert}~}v\in E}\right\}
,\end{align*}
and define
\({\mathcal{E}}\xrightarrow{\pi} X^Y = {\operatorname{Gr}}_k({\mathbb{C}}^n)\)
to be projection to the first factor such that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\pi^{-1}(E) \cong E \in {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}^{\dim = k}\)
  is a \(k{\hbox{-}}\)dimensional vector space for any \(E\in X^Y\).
\item
  \(\pi\) is \(G{\hbox{-}}\)equivariant:
  \(\pi(g\cdot (x, v)) = g \cdot \pi(x, v)\), where the first action is
  \(g\cdot (x, v) = (gx, gv)\), and \(\pi(x, v) = gx\). Moreover \(G\)
  acts on fibers linearly, so
  \(g\cdot({-}): \pi^{-1}(x) \to \pi^{-1}(gx)\) which sends \(E\to gE\)
  as subspaces in \({\mathbb{C}}^n\), and we require that this map of
  subspaces is a \({\mathbb{C}}{\hbox{-}}\)linear map.
\end{enumerate}

\end{definition}

\begin{remark}

Equivariant bundles on homogeneous spaces are determined by the
representation of the stabilizer on the corresponding fiber. We can pick
a base point
\({\operatorname{span}}_{\mathbb{C}}\left\{{e_1,\cdots, e_k}\right\} \cong {\mathbb{C}}^k \in {\operatorname{Gr}}_k({\mathbb{C}}^n)\),
whence \({\operatorname{Stab}}_G({\mathbb{C}}^k) = P\) is all but the
lower-left block:

\includegraphics{figures/2021-10-01_14-10-55.png}

Then \(\pi^{-1}({\mathbb{C}}^k) = {\mathbb{C}}^k\). We conclude
\begin{align*}
{\mathcal{E}}: G \overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}^k &\to G/P \\
[g, v] &\mapsto gv
.\end{align*}

\end{remark}

\begin{example}[?]

For \(k=1\), we're considering
\({\operatorname{Gr}}_1({\mathbb{C}}^n) = {\mathbb{P}}^{n-1}\).

\begin{itemize}
\item
  \(T \subseteq \operatorname{GL}_n\) are diagonal matrices, and
  \(t\curvearrowright{\left[ {x_1, 0, \cdots, 0} \right]} = {\left[ {tx_1, 0, \cdots, 0} \right]}\).
\item
  \(Y = \left\{{1\leq i \leq n-1 {~\mathrel{\Big\vert}~}{\left\langle {\lambda},~{\alpha_i {}^{ \vee }} \right\rangle} = 0}\right\} = \left\{{2, \cdots, n-1}\right\}\).
\item
  Taking \(\lambda = {\left[ {1, 0,\cdots, 0} \right]}\), we have a
  character
  \begin{align*}
  \lambda: T &\to {\mathbb{C}}^{\times}\\
  \operatorname{diag}(t_1,\cdots, t_n) &\mapsto t_1^1 t_2^0\cdots t_n^0
  .\end{align*}
\item
  \({\mathcal{E}}= G\overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}^1 = G\overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_{{\left[ {1, 0, \cdots, 0} \right]}} = {\mathcal{L}}(- \lambda)\).
\end{itemize}

Note that since this weight \(\lambda\) is dominant (and not
antidominant), there are no global sections.

\end{example}

\begin{remark}

Define
\begin{align*}
{\mathfrak{h}} {}^{ \vee }_{{\mathbb{Z}}, Y} \coloneqq\left\{{ \lambda\in {\mathfrak{h}} {}^{ \vee }_{\mathbb{Z}}{~\mathrel{\Big\vert}~}{\left\langle { \lambda},~{ \alpha_i  {}^{ \vee }} \right\rangle} = 0, i\in Y}\right\}
.\end{align*}
For any \(\lambda\in {\mathfrak{h}} {}^{ \vee }_{\mathbb{Z}}\) take
\(\lambda_1, \lambda_2 \in D_Y^0\) such that
\(\lambda= \lambda_1 - \lambda_2\), i.e.~we can write any weight as a
difference of dominant weights:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-10-01_14-21.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

Set
\begin{align*}
{\mathcal{L}}( \lambda) \coloneqq\mathcal{L} ^Y( - \lambda_2) \otimes{\mathcal{L}}( - \lambda_1) {}^{ \vee }
.\end{align*}
For example, given \(T \subseteq G\) and \(\lambda\in X(T)\), we have
\begin{align*}
\mathcal{L} (\lambda) = G\overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_{ - \lambda}
.\end{align*}

\end{remark}

\begin{remark}

Given \(w\in W\), define
\begin{align*}
{\mathcal{L}}_w( \lambda) \coloneqq P_{i_1}\overset{\scriptscriptstyle {B} }{\times} P_{i_2} \overset{\scriptscriptstyle {B} }{\times} \cdots \overset{\scriptscriptstyle {B} }{\times} P_{i_n} \overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ - \lambda}
.\end{align*}

\end{remark}

\begin{claim}

Let

\begin{itemize}
\tightlist
\item
  \({w} = (s_{i_1}, \cdots, s_{i_n})\)
\item
  \(i_{\lambda}: {\mathcal{G}}/P_Y \to {\mathbb{P}}(L^{\max} ( \lambda))\)
\item
  \(m_w: Z_w \to {\mathcal{G}}/P_Y\)
\end{itemize}

Then
\begin{align*}
{\mathcal{L}}_{w} (\lambda) = m_{w} {}^{ \vee }\mathcal{L} ^Y(\lambda)
.\end{align*}

\end{claim}

\begin{proof}[?]

Define
\begin{align*}
f: {\mathcal{L}}_{w}( \lambda) &\to Z_w 
= P_{i_1}\overset{\scriptscriptstyle {B} }{\times} P_{i_2} \overset{\scriptscriptstyle {B} }{\times} \cdots \overset{\scriptscriptstyle {B} }{\times} P_{i_n} \overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ - \lambda}
\\ 
[p_1,p_2, \cdots, p_n, z] &\mapsto [p_1, p_2, \cdots, p_n B/B] \\ \\ 
g: {\mathcal{L}}_{w}( \lambda) &\to {\mathcal{L}}^Y( \lambda)  \\
[p_1,p_2, \cdots, p_n, z] &\mapsto [p_1 \cdot p_2 \cdots p_n, z]
.\end{align*}

\begin{exercise}[?]

Check that these maps are well-defined.

\end{exercise}

Using the universal property of pullbacks, we get a diagram:

\begin{center}
\begin{tikzcd}
    {{\mathcal{L}}_w(\lambda)} \\
    \\
    && {m_w^* {\mathcal{L}}^Y(\lambda) = Z_w { \underset{\scriptscriptstyle {G/P} }{\times} } {\mathcal{L}}^Y(\lambda)} && {{\mathcal{L}}^Y(\lambda)} \\
    \\
    && {Z_w} && {G/P}
    \arrow["\pi", from=3-5, to=5-5]
    \arrow["{m_w}"', from=5-3, to=5-5]
    \arrow[from=3-3, to=5-3]
    \arrow[from=3-3, to=3-5]
    \arrow["g", curve={height=-24pt}, from=1-1, to=3-5]
    \arrow["f", curve={height=24pt}, from=1-1, to=5-3]
    \arrow["{\exists \varphi}", dashed, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMiw0LCJaX3ciXSxbNCw0LCJHL1AiXSxbNCwyLCJcXG1jbF5ZKFxcbGFtYmRhKSJdLFsyLDIsIm1fd14qIFxcbWNsXlkoXFxsYW1iZGEpID0gWl93IFxcZmliZXJwcm9ke0cvUH0gXFxtY2xeWShcXGxhbWJkYSkiXSxbMCwwLCJcXG1jbF93KFxcbGFtYmRhKSJdLFsyLDEsIlxccGkiXSxbMCwxLCJtX3ciLDJdLFszLDBdLFszLDJdLFs0LDIsImciLDAseyJjdXJ2ZSI6LTR9XSxbNCwwLCJmIiwwLHsiY3VydmUiOjR9XSxbNCwzLCJcXGV4aXN0cyBcXHZhcnBoaSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

The claim is that \(\phi\) is an isomorphism, we'll show this by
explicitly construction its inverse algebraic morphism. We have
\(\phi([p_1, \cdots, p_n, z]) = ( [p_1, \cdots, p_n B/B] \times [ p_1p_2\cdots p_n, z] )\).
Define
\begin{align*}
\psi: m_w^* {\mathcal{L}}^Y( \lambda) &\to {\mathcal{L}}_w( \lambda) \\
[p_1, \cdots, p_nB/B] \times [g, z] &\mapsto [p_1, \cdots, p_n, p_n^{-1}\cdots p_1^{-1}gz]
,\end{align*}
where \(p_1^{-1}\cdots p_1^{-1}g\in P\). This will clearly be an
inverse, it remains to show it's well-defined. Note that
\begin{align*}
p_1\cdots p_n P/P = gP/P \implies f^{-1}p_1\cdots p_n \in P
,\end{align*}
which follows from chasing the fiber product diagram around the two
sides.

\end{proof}

\begin{exercise}[?]

Check that this is well-defined by showing a different representative
has the same image. Then compose \(\phi, \psi\) in both orders.

\end{exercise}

\hypertarget{cohomology-of-certain-line-bundles-z_w-monday-october-04}{%
\section{\texorpdfstring{Cohomology of Certain Line Bundles \(Z_w\)
(Monday, October
04)}{Cohomology of Certain Line Bundles Z\_w (Monday, October 04)}}\label{cohomology-of-certain-line-bundles-z_w-monday-october-04}}

\begin{remark}

Some references:

\begin{itemize}
\item
  Fulton, \emph{Intersection Theory}. Similar difficulty to Hartshorne
  if you're going through it yourself!

  \begin{itemize}
  \tightlist
  \item
    See Young Tableaux books.
  \end{itemize}
\item
  Eisenbud-Harris, \emph{3264 and All That}. A more Vakil-style
  approach.
\end{itemize}

\end{remark}

\begin{definition}[Chow Group]

The \textbf{Chow group} of \(X\in{\mathsf{Var}}_{/ {k}}\) is the
quotient \(A_*(X) \coloneqq Z(X)/ \operatorname{Rat}(X)\), where
\(Z(X) = {\mathbb{Z}}[{\mathrm{Sub}}(X)]\), the free
\({\mathbb{Z}}{\hbox{-}}\)module on subvarieties of \(X\). The group
\(Z(X)\) are \textbf{algebraic cycle}, and we mod out by rational
equivalence.

\end{definition}

\begin{example}

If \(G\curvearrowright X\), then \(Y\sim gY\in A_*(X)\), and something
similar happens for many algebraic group actions. Another example is
that in \({\mathbb{P}}^1\), \(x \sim x'\) for all points \(x,x'\) since
\({\operatorname{PSL}}_2\curvearrowright{\mathbb{P}}^1\).

\end{example}

\begin{remark}

Note that there is also an equivariant Chow group/ring. In general,
\(A_*(X)\) is difficult/impossible to compute (according to Harris)
unless there is an affine stratification. In these cases, it coincides
with Borel-Moore homology.

\end{remark}

\begin{theorem}[?]

If \(X\) is smooth, then \(A^*(X)\) forms a ring, where the grading is
given by codimension of subvarieties. Thus there is a multiplication
\([A] \cdot [B] = [A \cap B]\) when \(A\pitchfork B\) generically. Here
transversality refers to an open condition on tangent spaces.

\end{theorem}

\begin{remark}

We have three ways of thinking about line bundles:

\begin{itemize}
\tightlist
\item
  Local trivializations
\item
  Algebraic morphisms with 1-dimensional fibers
\item
  Invertible sheaves
\end{itemize}

Now we'll add a fourth in terms of divisors. Define:

\begin{itemize}
\tightlist
\item
  \(A_{n-1}(X)\in {\mathsf{Grp}}\), \textbf{Weil divisors}
\item
  \({\operatorname{Pic}}(X) \in {\mathsf{Grp}}\), the group of
  isomorphism classes of algebraic line bundles on \(X\) where
  \([L_1] \cdot [L_2] \coloneqq[L_1 \oplus L_2]\).
\end{itemize}

\end{remark}

\begin{proposition}[?]

Taking the Chern class yields a group morphism
\(c_1: {\operatorname{Pic}}(X) \to A_{n-1}(X)\). If the line bundle is
generated by global sections, take the zero section of the global
section. If \(X\) is smooth, \(c_1\) is an isomorphism, and we write
\(c_1({\mathcal{O}}_X(Y)) \coloneqq[Y]\in A_{n-1}(X)\). Note that this
is slightly different to the ideal sheaf definition in Vakil.

\end{proposition}

\begin{remark}

See relation to Schubert varieties and Grassmannians in the referenced
books. Bott-Samelson-Demazure and flag varieties will be smooth,
although we'll have to be careful for Schubert varieties.

\end{remark}

\begin{proposition}[8.1.2]

Define the length of a word \(w\in W\) to be the number of simple
reflections, regardless of whether or not \(w\) is reduced. Let
\(n\coloneqq\ell(w)\), then there is a formula for the canonical bundle
\(K_{Z_w}\) of any Bott-Samelson-Demazure variety \(Z_w\) (even
Kac-Moody types):
\begin{align*}
{\mathcal{L}}_w(-\rho) \otimes{\mathcal{O}}_{Z_w}( - \sum_{q=1}^n Z_{w(q)} )
.\end{align*}

\end{proposition}

\begin{remark}

Here \(\rho\in {\mathfrak{h}} {}^{ \vee }_{\mathbb{Z}}\)
(e.g.~characters of the torus in the semisimple simply connected case)
is any element satisfying \(\rho( \alpha_i {}^{ \vee }) = 1\) for all
\(1\leq i \leq \ell\). Recall that
\begin{align*}
Z_w = P_{i_1}\overset{\scriptscriptstyle {B} }{\times} \cdots \overset{\scriptscriptstyle {B} }{\times} P_{i_n} / B = \left\{{ {\left[ {p_1, \cdots, p_n B/B } \right]}}\right\}
,\end{align*}
and \(Z_w(q)\) means deleting the \(q\)th factor, so
\(Z_w(q) = \left\{{{\left[ {p_1,\cdots, 1,\cdots, p_nB/B} \right]}}\right\}\)
has the \(q\)th coordinate set to 1. Note that there is a quotient map
\(Z_w \to Z_{w(n)}\), which has a section, and we can use this to
induct.

\end{remark}

\begin{proof}[?]

Consider \(G\) connected and reductive and let \(X=G/B\) be the flag
variety, which is smooth. Then for \(\lambda \in X(T)\) corresponds to
the algebraic line bundle
\({\mathcal{L}}^{\emptyset}( \lambda) = G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ - \lambda}\).
This yields a function
\(X(T) \to {\operatorname{Pic}}(X) \xrightarrow{c_1} A_{n-1}(X)\) given
by forgetting the \(G{\hbox{-}}\)action. This is a group morphism, where
adding characters maps to tensoring bundles.

Note that for \(T = {\mathbb{C}}^{\times}\), we have
\begin{align*}
X(T) = \mathop{\mathrm{Hom}}_{{\mathsf{Alg}}{\mathsf{Grp}}}({\mathbb{C}}^{\times}, {\mathbb{C}}^{\times})= \left\{{z \mapsto z^k {~\mathrel{\Big\vert}~}k\in {\mathbb{Z}}}\right\} { { \, \xrightarrow{\sim}\, }}_{{\mathsf{Ab}}{\mathsf{Grp}}} {\mathbb{Z}}
,\end{align*}
where negatives are permitted since \(0\not\in {\mathbb{C}}^{\times}\).
More generally,
\(X(T) { { \, \xrightarrow{\sim}\, }}_{{\mathsf{Ab}}{\mathsf{Grp}}} {\mathbb{Z}}^n\)
for \(n=\operatorname{rank}T\), where
\({\left[ {{ {t}_1, {t}_2, \cdots, {t}_{n}}} \right]} \xrightarrow{\lambda} \lambda_1^{k_1}\cdots \lambda_n^{k_n}\).
Since we have an affine stratification by Schubert cells, we can write
\(A_*(X) = \bigoplus_{w\in W} [X_w]\), and in fact
\(A_k(X) = \bigoplus _{\ell(w) = k} [X_w]\). Considering the lattice for
\(W\), there are \(\ell\) dimension 1 Schubert cells, and identifying
them as CW cells and applying Poincare duality, there are \(\ell\)
codimension 1 cells:

\begin{center}
\begin{tikzcd}
    && w \\
    \\
    {w_0 s_1} && {w_0 s_j} && {w_0 s_n} \\
    \vdots && \vdots && \vdots \\
    \vdots && \vdots && \vdots \\
    {s_1} & \cdots & {s_j} & \cdots & {s_n} \\
    \\
    && e
    \arrow[from=3-1, to=1-3]
    \arrow[from=3-3, to=1-3]
    \arrow[from=3-5, to=1-3]
    \arrow[from=8-3, to=6-1]
    \arrow[from=8-3, to=6-3]
    \arrow[from=8-3, to=6-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTYsWzIsNywiZSJdLFswLDUsInNfMSJdLFsyLDUsInNfaiJdLFs0LDUsInNfbiJdLFsxLDUsIlxcY2RvdHMiXSxbMyw1LCJcXGNkb3RzIl0sWzAsMiwid18wIHNfMSJdLFsyLDIsIndfMCBzX2oiXSxbNCwyLCJ3XzAgc19uIl0sWzIsMCwidyJdLFsyLDMsIlxcdmRvdHMiXSxbMiw0LCJcXHZkb3RzIl0sWzAsNCwiXFx2ZG90cyJdLFswLDMsIlxcdmRvdHMiXSxbNCwzLCJcXHZkb3RzIl0sWzQsNCwiXFx2ZG90cyJdLFs2LDldLFs3LDldLFs4LDldLFswLDFdLFswLDJdLFswLDNdXQ==}{Link
to Diagram}
\end{quote}

It turns out that the map is given as follows:
\begin{align*}
{\mathbb{Z}}^n \cong X(T) &\longrightarrow A_{n-1}(X) \cong {\mathbb{Z}}^\ell \\
\lambda&\mapsto \sum_{i=1}^\ell {\left\langle {\lambda},~{ \alpha_i {}^{ \vee }} \right\rangle}[X_{w_0 s_i}] && n\geq \ell
.\end{align*}

\end{proof}

\begin{example}[?]

For
\(G = {\operatorname{SL}}_2, {\mathcal{L}}( \lambda_k) = {\mathcal{O}}_{{\mathbb{P}}^1}(k)\)
and \(X(T) \cong {\mathbb{Z}}\). Recall that
\({{\Gamma}\qty{{\mathbb{P}}^1, {\mathcal{O}}_{{\mathbb{P}}^1}(k)} } = {\mathbb{C}}[x, y]_k\)
are homogeneous polynomials of degree \(k\) when \(k\geq 0\), otherwise
there are no global sections. For example,
\({\mathbb{C}}[x, y]_2 = \left\langle{x^2, xy, y^2}\right\rangle\) is
dimension \(3 = 2 + 1\). All points are rationally equivalent, so we can
take the basepoint \(B/B\), and so the map will need to track the
multiplicity of points. The composition is given by the following:

\begin{center}
\begin{tikzcd}
    {X(T)} && {{\operatorname{Pic}}(X)} && {A_{n-1}(X)} \\
    \\
    {{\mathcal{L}}(\lambda_k)} && {{\mathcal{O}}_{{\mathbb{P}}^1}(k)} && {k[B/B]}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[maps to, from=3-1, to=3-3]
    \arrow[maps to, from=3-3, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCJYKFQpIl0sWzIsMCwiXFxQaWMoWCkiXSxbNCwwLCJBX3tuLTF9KFgpIl0sWzIsMiwiXFxPT197XFxQUF4xfShrKSJdLFs0LDIsImtbQi9CXSJdLFswLDIsIlxcbWNsKFxcbGFtYmRhX2spIl0sWzAsMV0sWzEsMl0sWzUsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFszLDQsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=}{Link
to Diagram}
\end{quote}

The cotangent bundle of \(X\) is given by
\(G\overset{\scriptscriptstyle {P} }{\times} u = T {}^{ \vee }G/P\)
where \(P = LU\). The canonical bundle is the top wedge power, and here
we get
\(G\overset{\scriptscriptstyle {B} }{\times} {\mathfrak{n}}= G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_2 = {\mathcal{L}}(-2)\),
noting that the canonical is equal to the cotangent bundle here, and
we've identified which equivariant bundle this is.

\end{example}

\hypertarget{friday-october-08}{%
\section{Friday, October 08}\label{friday-october-08}}

\begin{remark}

Continuing some stuff from Kumar Ch. 8: the goal is to understand the
Demazure and Weyl-Kac character formulas. Open question: how can one
compute the singular locus of a given Schubert variety? This is
surprisingly a hot topic this semester, c/o multiple Arxiv papers that
have come out over the past few months.

Our first goal: showing \(X_w^Y\) is normal. Note that most varieties in
representation theory are not normal, and this complicates things
significantly, so normality is a great condition here.

Recall that for \(X\in {\mathsf{Var}}\), the stalks
\({\mathcal{O}}_{X, x}\) are local rings, and the \textbf{cotangent
space at \(x\)} is defined as \({\mathfrak{m}}_x/{\mathfrak{m}}_x^2\).

\begin{quote}
Cohomology vanishing: some of the hardest and most important results in
this area!
\end{quote}

\end{remark}

\begin{theorem}[8.1.8, Main Result]

Let \(w = (s_{i_1} \cdots, s_{i_n}) \in W\) be a word and consider
\(j, k\) such that \(1\leq j\leq k\leq n\). Suppose that the subword
\(v = (s_{i_j} \cdots, s_{i_k})\) is reduced. Considering the associated
BSDH-varieties, we have a subvariety
\begin{align*}
Z_v \coloneqq P_{i_j}\overset{\scriptscriptstyle {B} }{\times} \cdots P_{i_k}/B \hookrightarrow
Z_w \coloneqq P_{i_1}\overset{\scriptscriptstyle {B} }{\times} \cdots P_{i_n}/B
.\end{align*}
Recall that
\({\mathcal{L}}^Y(\lambda) \coloneqq G\overset{\scriptscriptstyle {P_Y} }{\times} {\mathbb{C}}_{- \lambda}\),
and
\begin{align*}
{\mathcal{L}}_w( \lambda) \coloneqq P_{i_1}\overset{\scriptscriptstyle {B} }{\times} \cdots \overset{\scriptscriptstyle {B} }{\times} P_{i_n} {\mathbb{C}}_{- \lambda} = f^* {\mathcal{L}}^Y( \lambda)
,\end{align*}
and we write \(w(n)\) for \(w\) with the \(n\)th letter omitted.
Moreover codimension 1 subvarieties correspond to line bundles under the
Chern class isomorphism. Then for any integral dominant
\(\lambda \in D_{\mathbb{Z}}\), there are 3 vanishing formulas:

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

  \begin{align*}
  H^{\geq 1 } \qty{ Z_w; {\mathcal{L}}_w( \lambda) \otimes{\mathcal{O}}_{Z_w}(- \sum_{q=0}^k Z_{w(q)} ) } &= 0 
  .\end{align*}
\item

  \begin{align*}
  H^{\geq 1} \qty{Z_w; {\mathcal{L}}_w( \lambda) } &= 0 
  .\end{align*}
\item
  If \(k<n\) and \(v' \coloneqq(s_{i_k} \cdots, s_{i_k} s_{i_{k+1}})\)
  is \emph{not} reduced, then
\end{enumerate}

\begin{align*}
H^{\geq 0 } \qty{ Z_w; {\mathcal{L}}_w( \lambda) \otimes{\mathcal{O}}_{Z_v}(- \sum_{q=j}^k Z_{w(q)} ) } &= 0 \\
.\end{align*}

\end{theorem}

\begin{remark}

We'll often use \textbf{Serre duality} in the following form: given a
set of nice assumptions, there is a perfect pairing
\begin{align*}
H^i(X; {\mathcal{F}}) \times H^{n-i}(X; K_X \otimes{\mathcal{F}} {}^{ \vee }) \to {\mathbb{C}}
,\end{align*}
where
\({\mathcal{F}} {}^{ \vee }\coloneqq\mathop{\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{F}}, {\mathcal{O}}_X)\)
and \(K_X\) is the dualizing sheaf. Note that if \(X\) is smooth and
projective, one can take \(K_X\) to be the canonical sheaf.

\end{remark}

\begin{lemma}[8.18]

For any finite-dimensional representation \(M\) of
\(B{ {}^{ \scriptscriptstyle\times^{n} } }\), there is a functorial
assignment a \(P_{i_1}{\hbox{-}}\)equivariant algebraic vector bundle
\({\mathcal{L}}_w(M) \to Z_w\) on the BSDH variety. which is an exact
functor on
\({\mathsf{B{ {}^{ \scriptscriptstyle\times^{n} } }}{\hbox{-}}\mathsf{Mod}}^{\dim < \infty}\),
given by
\begin{align*}
{\mathcal{L}}_w( - \lambda ) = {\mathcal{L}}_w( {\mathbb{C}}_{ \lambda})
.\end{align*}
This is induced by
\(B{ {}^{ \scriptscriptstyle\times^{n} } } \xrightarrow{{\operatorname{pr}}_n} B \to {\mathbb{C}}_{\lambda}\).

\end{lemma}

\begin{remark}

Using this formula,
\begin{align*}
{\mathcal{L}}(\lambda) {}^{ \vee }= {\mathcal{L}}( {\mathbb{C}}_{- \lambda}) {}^{ \vee }= {\mathcal{L}}({\mathbb{C}}_{- \lambda}  {}^{ \vee })
,\end{align*}
where given \(V_{ \lambda}\) a highest weight representation of \(G\)
(connected reductive finite type), we have
\(V_{ \lambda} {}^{ \vee }= V_{-w_0 \lambda}\). Here
\({\mathbb{C}}_{ - \lambda}\) is a representation of the torus, for
which \(w_0 = \operatorname{id}\).

\end{remark}

\begin{example}[?]

For \(w = v = (s)\) and \(1\leq j\leq k \leq n =1\), we have
\(Z_{(s)} = P_s/B \cong {\mathbb{P}}^1\). Noting
\(Z_{ \emptyset} = B/B = {\operatorname{pt}}\), using the formula we
obtain
\begin{align*}
H^p( P_s/B; {\mathcal{L}}_{(s)} ( \lambda) \otimes{\mathcal{O}}_{Z_{(s)}} (-B/B) )
.\end{align*}
This is an (equivariant) line bundle on \({\mathbb{P}}^1\), which are
all of the form \({\mathcal{O}}(n)\) -- which one is it? Write
\(a \coloneqq{\left\langle { \lambda},~{ \alpha {}^{ \vee }} \right\rangle} \in {\mathbb{Z}}_{\geq 0}\)
since \(\lambda\) was dominant integral. Forgetting the group action
yields an algebraic bundle which we can write as
\begin{align*}
H^p(P_s/B ; {\mathcal{O}}_{{\mathbb{P}}^1}(a) \otimes{\mathcal{O}}_{{\mathbb{P}}^1}(-1) )
.\end{align*}

This can also be described by tensoring
\({\mathbb{C}}_{ \lambda} \otimes{\mathbb{C}}_{ \mu} \cong {\mathbb{C}}_{ \lambda+ \mu}\).
Finally, we can identify this homology with
\begin{align*}
H^p(P_s/B ; {\mathcal{O}}_{{\mathbb{P}}^1}(a-1) )
.\end{align*}

Note that the canonical for \({\mathbb{P}}^1\) is
\(G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{2} = {\mathcal{L}}(-2) = {\mathcal{O}}(-2)\)
(noting that \(-n\) has no global sections). So if
\({\mathcal{F}}= {\mathcal{O}}(k)\) then
\({\mathcal{F}} {}^{ \vee }= {\mathcal{O}}(-k)\). Applying Serre duality
yields
\begin{align*}
H^i( {\mathbb{P}}^1; {\mathcal{O}}(k) ) \times H^{n-i}({\mathbb{P}}^1; {\mathcal{O}}(-2) \otimes{\mathcal{O}}(-k) ) \to {\mathbb{C}}
.\end{align*}

Note that \(H^0({\mathbb{P}}^1; {\mathcal{O}}(k)) = 0\) for \(k<0\),
since these look like homogeneous polynomials in degree \(k\) (and there
are none of negative degree), so taking \(k=-1\) we have
\(H^0({\mathbb{P}}^1; {\mathcal{O}}(-1)) = 0\). By duality, this pairs
with \(H^1({\mathbb{P}}^1; {\mathcal{O}}(-1))\), and continuing yields a
pairing:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-10-08_14-34.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{example}

\begin{example}[?]

Let \(w=(s, s), v=(s), v'=(s), \eta = \emptyset, \mu = \emptyset\).
Write \(\sigma = \psi: Z_w \to Z_v\), which is projection onto the first
coordinates in the corresponding BSDH varieties:
\begin{align*}
P_s \overset{\scriptscriptstyle {B} }{\times} P_s/B &\to P_s/B \\
[p_1, p_2B/B] &\mapsto [p_1 B/B]
.\end{align*}
Note that
\begin{align*}
{\mathcal{O}}_{Z_w}( -B\overset{\scriptscriptstyle {B} }{\times} P_s /B) = \sigma^* ( {\mathcal{O}}_{Z_v}(-B/B) )
.\end{align*}
There are 3 important facts we'll revisit:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  A projection formula,
\item
  Lemma 8.1.5,
\item
  The Leray spectral sequence.
\end{enumerate}

\end{example}

\hypertarget{monday-october-11}{%
\section{Monday, October 11}\label{monday-october-11}}

\begin{remark}

Chapter 8: actually equivariant \({\mathsf{K}}{\hbox{-}}\)theory without
saying so! Also deals with Demazure operators. Goal: show that \(X^Y_w\)
is normal.

\end{remark}

\begin{definition}[Normal varieties]

Let \(X\in {\mathsf{Var}}_{/ {k}}\) be irreducible, then \(X\) is
\textbf{normal} at \(x\in X\) iff
\({\mathfrak{m}}_x \coloneqq{\mathcal{O}}_{X, x}\) is integrally closed
in its field of fractions \(\operatorname{ff}({\mathfrak{m}}_x)\).

\end{definition}

\begin{remark}

Note that there are implications smooth \(\implies\) factorial
\(\implies\) normal in \({\mathsf{Var}}_{/ {k}}\). We write
\(\Sigma(X) \subseteq X\) to be the singular locus, and if \(X\) is
normal then \(\operatorname{codim}_X \Sigma(X) \leq 2\).

\end{remark}

\begin{example}[Whitney's Umbrella]

Let \(f(x,y,z) = x^2-zy^2\) and consider
\(X \coloneqq V(f) \subseteq {\mathbb{A}}^3_{/ {{\mathbb{C}}}} \in {\mathsf{Aff}}{\mathsf{Var}}_{/ {{\mathbb{C}}}}\).
Checking normality for affine varieties just amounts to checking on
regular functions, so \(X\) is normal iff
\({\mathbb{C}}[X] \hookrightarrow{\mathbb{C}}(X)\) is integrally closed.
One direction involves checking that localizations are integrally
closed, which is an easy exercise in commutative algebra, while the
other direction is harder. Consider \(X({\mathbb{R}})\):

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-10-11_14-03.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

The claim is that \(X\) is not normal. Setting \(\xi = x/y\) is not a
regular function on \(X\) since \(y\) vanishes at some points of \(X\),
but \(\xi^2 = x^2/y^2 = z \in {\mathbb{C}}[X]\) is regular.

\end{example}

\begin{remark}

Motivating question: normality is a local condition, so where can \(X\)
be non-normal? There is a process of \textbf{normalization} which
associates to \(X\) a unique normal variety \(\tilde X\) with a unique
finite birational morphism \(\nu: \tilde X\to X\). Here \emph{finite}
means points have finite fibers and the map is proper.

Some properties: - \(\nu\) is unique.

\begin{itemize}
\tightlist
\item
  \(\tilde X \xrightarrow{\nu} X\) satisfies a universal property: For
  \(X\to Y\) for any \(Y\) normal, then there exists a unique lift
\end{itemize}

\begin{center}
\begin{tikzcd}
    {\tilde X} \\
    \\
    X && Y
    \arrow["f", from=3-1, to=3-3]
    \arrow["\nu", from=3-1, to=1-1]
    \arrow["{\exists ! \tilde f}", dotted, from=1-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwyLCJYIl0sWzIsMiwiWSJdLFswLDAsIlxcdGlsZGUgWCJdLFswLDEsImYiXSxbMCwyLCJcXG51Il0sWzIsMSwiXFxleGlzdHMgISBcXHRpbGRlIGYiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkb3R0ZWQifX19XV0=}{Link
to Diagram}
\end{quote}

\begin{quote}
See also \textbf{Stein factorization} for proper morphisms.
\end{quote}

\begin{itemize}
\item
  If \(f:X\to Y\) is a birational projective morphism between
  irreducible varieties and \(Y\) is normal, then
  \(f_* {\mathcal{O}}_X = {\mathcal{O}}_Y\).

  \begin{quote}
  See also \textbf{Zariski's main theorem}.
  \end{quote}
\end{itemize}

\end{remark}

\begin{example}[?]

Let \(X\) be the umbrella from above. Consider
\(\nu(u, v) = {\left[ {uv, v, v^2} \right]}\), so
\({\mathbb{A}}^2_{/ {{\mathbb{C}}}} \xrightarrow{\nu} X \subseteq {\mathbb{A}}^3_{/ {{\mathbb{C}}}}\),
and let \(f(x,y,z) = x^2 - zy^2\), so
\(f(uv, v, v^w) = (uv)^2 - u^2v^2 = 0\) is a regular function on \(X\).
One can check that \(\Im(\nu) \subseteq X\) so this is surjective, and
the conclusion is that \(X\) is irreducible with 2-dimensional fibers.
Consider the fibers of \(\nu\):

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(\mathbf{x} = 0\) yields
  \(\nu^{-1}(\mathbf{x}) = \left\{{ {\left[ {u, v} \right]} \in {\mathbb{A}}^2 {~\mathrel{\Big\vert}~}{\left[ {uv, v, u^2} \right]} = \mathbf{0}}\right\} = {\operatorname{pt}}\).
\item
  \(\mathbf{x} = {\left[ {0,0, z} \right]}\) with \(z\neq 0\) yields
  \(\nu^{-1}(\mathbf{x}) = \left\{{{\left[ {uv, v, u^2} \right]} = {\left[ {0,0,z} \right]}}\right\} = \left\{{p_1, p_2}\right\}\)
  which have nonzero 2nd coordinates, by choosing a square root of
  \(u\).
\item
  \(\mathbf{x} = {\left[ {x,y,z} \right]}\) with \(x\neq 0\) yields
  \(\nu^{-1}(\mathbf{x}) = \left\{{{\left[ {uv, v, u^2} \right]} = {\left[ {x,y,z} \right]}}\right\}\).
  This forces \(v=y\), and \(x = uv= uy\) which is nonzero and can be
  solved for \(u\), so we again get a single point
  \(\nu^{-1}(\mathbf{x}) = {\operatorname{pt}}\).
\end{enumerate}

Note that just considering the real points misses the entire \(-z\)
axis. This can be analyzed by regarding \(u,v\in {\mathbb{C}}\) as a
pair of points in the same plane; then if \(u=v=0\) corresponds to (1),
\(v=0\) with \(u\) varying yields (2) (and two-point fibers), and moving
\(v\) from 0 yields (3). Here \(X\) is normal at the points in (1), but
not normal in (2) and (3).

Moral: we can study singularities by looking at fibers.

\end{example}

\begin{remark}

Next time: Schubert varieties.

\end{remark}

\hypertarget{wednesday-october-13}{%
\section{Wednesday, October 13}\label{wednesday-october-13}}

\begin{remark}

Goal: show Schubert varieties are normal.

\end{remark}

\begin{theorem}[8.2.2]

Let \(v\leq w\in W\),
\(\lambda \in D_{\mathbb{Z}}\cap{\mathfrak{h}} {}^{ \vee }_{{\mathbb{Z}}, Y}\)
where we take the extension
\(P_Y \xrightarrow{\lambda} {\mathbb{C}}^{\times}\) to the parabolic.
Then part (b) of the theorem states that \(X_W^Y\) is normal.

\end{theorem}

\begin{proof}[?]

Let \(w\in W_Y'\) such that \(w'\) is a minimal length representative in
\(w W_Y\). Write \(\pi(w') = w\) for the element obtained by multiplying
the elements in the word \(w'\), and choose a word
\(\mathcal{w} \in \mathcal{W}\) such that \(\pi \mathcal{w}' = w'\).
Then \(m_{{\mathcal{W}}}^Y: Z_{{\mathcal{W}}'} \to X_{w'}^Y\) is
surjective and birational, and so the following induced hom is an
isomorphism
\begin{align*}
(m_{{\mathcal{W}}}^Y)^*: H^0(X_W^Y, {\mathcal{L}}_w^Y( \lambda)) { { \, \xrightarrow{\sim}\, }}H^0(Z_{w'}, {\mathcal{L}}_{w'} ( \lambda)) 
.\end{align*}
Taking any \(\lambda^0 \in D^0_Y\) and applying A.32 (a deep AG fact) to
the ample line bundle \({\mathcal{L}}= {\mathcal{L}}_W^Y(\lambda_0)\),
we get the following important formula:

\begin{align*}
(m_{w'}^Y)_* {\mathcal{O}}_{Z_{w'}} = {\mathcal{O}}_{X_W^Y}
.\end{align*}
This is what Kumar spends most of the time showing, and is essentially
equivalent to the following:

\begin{fact}[Zariski's Main Theorem]

Let \(f:X\to Y\) be birational and proper such that \(X\) is normal.
Then \(Y\) is normal iff \(f_* {\mathcal{O}}_X = {\mathcal{O}}_Y\),
which implies that the fibers are connected. This is proved in
Hartshorne.

\end{fact}

\begin{quote}
Vogan: there are more statements in representation theory that say
``\textbf{if} normal'' than there are that say ``\textbf{then} normal''.
\end{quote}

Recall that the normalization \(\tilde Y \xrightarrow{\nu} Y\) satisfies
a universal property with respect to maps from normal varieties. Using
functoriality, we have
\begin{align*}
f_* {\mathcal{O}}_X 
&= (\nu \circ \tilde f)_* {\mathcal{O}}_X \\
&= \nu_* (\tilde f_* {\mathcal{O}}_X) \\
&= \nu_* {\mathcal{O}}_{\tilde Y} &&\text{Zariski's Main Theorem, forward direction}\\
&= {\mathcal{O}}_Y &&\text{by assumption}
.\end{align*}
Use that \(\tilde f\) is birational and proper, where properness can be
shown by exhibiting it as the pullback of a proper morphism. Using that
\(Y\) is normal iff every open affine \(U \subseteq Y\) is normal, we
have
\begin{align*}
{\mathcal{O}}_Y(U) = (\nu_* {\mathcal{O}}_{\tilde Y})(U) = {\mathcal{O}}_{\tilde Y}(\nu^{-1}(U))
.\end{align*}

\end{proof}

\hypertarget{borel-weil-homomorphism}{%
\subsection{Borel-Weil Homomorphism}\label{borel-weil-homomorphism}}

\begin{remark}

For any \(V\in {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) with
\(\dim_{\mathbb{C}}\leq \infty\), define a morphism
\begin{align*}
\beta_V: V {}^{ \vee }&\to H^0( {\mathbb{P}}V, {\mathcal{L}}_V {}^{ \vee }) \\
f &\mapsto (\delta \mapsto (\delta, { \left.{{f}} \right|_{{\delta }} }))
,\end{align*}
where taking the dual of the tautological amounts to, for each line
\(\delta \in {\mathbb{P}}V\), quotienting by the annihilator to get
\(V {}^{ \vee }/ \delta^\perp\). Note that there is a projection
\(\pi: {\mathcal{L}}_V {}^{ \vee }\to {\mathbb{P}}V\).

Take \(\lambda \in D_{\mathbb{Z}}\) and define a morphism of
\({\mathsf{G}{\hbox{-}}\mathsf{Mod}}\)
\begin{align*}
\beta = \beta(\lambda): L^{\max}( \lambda) {}^{ \vee }\to H^0({\mathcal{X}}, {\mathcal{L}}( \lambda))
,\end{align*}
where \({\mathcal{X}}\) denotes that this works in the Kac-Moody
setting. Note that \({\mathcal{G}}\) acts naturally on
\({\mathcal{L}}^Y(\lambda)\) and thus on
\(H^p({\mathcal{X}}^Y, {\mathcal{L}}^Y( \lambda))\), and recall
\(G\overset{\scriptscriptstyle {P_Y} }{\times}{\mathbb{C}}_{- \lambda} \to G/P_Y = X^Y\).
Then \(X_w \subseteq X\) and
\(\beta_w( \lambda): L^{\max}( \lambda) {}^{ \vee }\to H^0(X_w, {\mathcal{L}}_w( \lambda))\)

\end{remark}

\begin{remark}

How does this relate to representation theory? Let \(V\) be an
irreducible integrable \({\mathfrak{g}}{\hbox{-}}\)module with highest
weight \(\lambda\), then every \(w\in W\) induces \(V_w\), and
\(U({\mathfrak{b}}){\hbox{-}}\)submodule generated by extremely weight
vectors \(w_{w \lambda}\). Then \(\beta\) acts by pushing weights
``up'', and so e.g.~if one has weights
\(\lambda, w_1 \lambda, w_2 \lambda, \cdots\) one can consider the
\textbf{Demazure submodule} generated by any given \(w_i \lambda\).
Often we set \(V = L^{\max}(\lambda)\), and so
\begin{align*}
(L^{\max} (\lambda))_w = L_w^{\max}( \lambda)
.\end{align*}

\end{remark}

\begin{remark}

Going back to part (a) of the theorem, we have isomorphisms:

\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5mu\beta\mkern-1.5mu}\mkern 1.5mu_w^Y: L_w^{\max}( \lambda) {}^{ \vee }&{ { \, \xrightarrow{\sim}\, }}H^0( X_w^Y; {\mathcal{L}}_w^Y(\lambda)) \\
\alpha {}^{ \vee }: H^0( X_w^Y; {\mathcal{L}}_w^Y( \lambda)) &{ { \, \xrightarrow{\sim}\, }}H^0(X_w, {\mathcal{L}}_w( \lambda))
.\end{align*}
We have the following geometric picture:

\begin{center}
\begin{tikzcd}
    {G_w \coloneqq\mkern 1.5mu\overline{\mkern-1.5muBwB\mkern-1.5mu}\mkern 1.5mu} && G \\
    \\
    {X_w} && {G/B} \\
    \\
    {X_w^Y} && {G/P}
    \arrow[from=1-1, to=3-1]
    \arrow[from=3-1, to=5-1]
    \arrow[from=1-3, to=3-3]
    \arrow[from=3-3, to=5-3]
    \arrow["\subseteq", hook, from=5-1, to=5-3]
    \arrow["\subseteq", hook, from=3-1, to=3-3]
    \arrow["\subseteq", hook, from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCJHX3cgXFxkYSBcXGJhcntCd0J9Il0sWzIsMCwiRyJdLFsyLDIsIkcvQiJdLFsyLDQsIkcvUCJdLFswLDIsIlhfdyJdLFswLDQsIlhfd15ZIl0sWzAsNF0sWzQsNV0sWzEsMl0sWzIsM10sWzUsMywiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzQsMiwiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzAsMSwiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV1d}{Link
to Diagram}
\end{quote}

The connection between representation theory and geometry is th
following:
\begin{align*}
H^0(Z_\infty; {\mathcal{L}}_\infty ( \lambda ) ) { { \, \xrightarrow{\sim}\, }}L^{\max}( \lambda) {}^{ \vee }
.\end{align*}

\end{remark}

\begin{remark}

These statements are easy to remember and use but hard to prove! So
we'll move on and look at the Demazure character formula.

\end{remark}

\hypertarget{ch.8-continued-monday-october-18}{%
\section{Ch.8 Continued (Monday, October
18)}\label{ch.8-continued-monday-october-18}}

\begin{remark}

Today: looking at more examples of Schubert varieties in detail,
e.g.~\({\mathsf{Sp}}_{2n}\). One can take
\(G' \coloneqq\operatorname{GL}_{2n}\) and define involutions
\(G' \xrightarrow{\Theta} G'\). One example is \(g\mapsto g^{-t}\),
whose fixed points are \({\operatorname{O}}_{2n}\), and it's easy to
check that this is an involution:
\begin{align*}
(\Theta')^2(g) = \Theta' ({}^t g^{-1}) = {}^t( {}^t g^{-1})^{-1}= {}^t ({}^t g) = g
.\end{align*}
For \({\mathsf{Sp}}_{2n}\), taking
\begin{align*}
\theta(g) = -J^t g J 
\end{align*}
where \(J\) is the matrix
\begin{align*}
\begin{bmatrix}
 &  &  &  &  &  & 1 \\
 &  &  &  &  & \cdots &  \\
 &  &  &  & 1 &  &  \\
 &  &  & -+- &  &  &  \\
 &  & -1 &  &  &  &  \\
 & \cdots &  &  &  &  &  \\
-1 &  &  &  &  &  &  
\end{bmatrix}
.\end{align*}

We can check that this is an involution:
\begin{align*}
\Theta^2(g) 
&= \Theta(-J^t g J) \\
&= -J^t(-J^t g J)^{-1}J \\
&= J(Jg^{-t} J)^{-1}J \\
&= JJ g JJ \\
&= g
.\end{align*}

\end{remark}

\begin{definition}[?]

\((G')^\Theta \coloneqq\left\{{g'\in G'{~\mathrel{\Big\vert}~}\Theta(g') = g'}\right\}\)
are the fixed points under the involution \(\Theta\).

\end{definition}

\begin{proposition}[?]

One can write
\begin{align*}
(G')^\Theta = \left\{{g\in G' {~\mathrel{\Big\vert}~}\omega(g' x, g' y) = \omega(x, y)}\right\}
.\end{align*}
for \(\omega\) the associated bilinear form
\(\omega(x, y) = { {}^t{x}} J y\). Note that \(g'x, g'y\) should be
column vectors here.

\end{proposition}

\begin{proof}[sketch]

Write the RHS set as
\(\left\{{g\in G {~\mathrel{\Big\vert}~}{ {}^t{g'}} J g' = J}\right\}\).
Then check that if \(\theta(g) = g\) for some \(g\in G'\),
\begin{align*}
\omega(gx, gy) 
&= { {}^t{(gx)}}  J (gy) \\
&= { {}^t{(x^{-1})}}  g J  gy \\
&= { {}^t{(x^{-1})}}  g J  \Theta(g) y \\
&= { {}^t{(x^{-1})}}  g J (-J { {}^t{g^{-1}}}  J ) y \\
&= { {}^t{x}}  Jy
.\end{align*}
So these two act the same on all elements \(x, y\), and thus have the
same matrix, yielding \(\subseteq\). For the reverse containment, if
\(\omega(gx, gy) = \omega(x, y)\), then
\begin{align*}
{ {}^t{g}}  J g &= J \\
\implies Jg &= { {}^t{g^{-1}}}  J \\
\implies \Theta(g) &= -J { {}^t{g^{-1}}}  J \\
&= -JJ g \\
&= g
.\end{align*}

\end{proof}

\begin{remark}

We can realize \({\mathsf{Sp}}_{2n}\) as \((G')^{\Theta}\).

\end{remark}

\begin{fact}

How do we get a Borel? It is a general fact that these can be obtained
by intersecting with Borels in the ambient group, so take
\(B' \cap{\mathsf{Sp}}_{2n}\) for \(B' \subseteq G'\) upper triangular.
Then \(B'\) is \(\Theta{\hbox{-}}\)stable:

\includegraphics{figures/2021-10-18_14-25-12.png}

\end{fact}

\begin{remark}

Let \(G = (G')^\Theta\), then \(G\curvearrowright G'/B'\) with finitely
many orbits. So we get closure relations:

\includegraphics{figures/2021-10-18_14-27-13.png}

One can also fix \(T' \subseteq G'\) as a maximal torus of diagonal
matrices, and this is also \(\Theta{\hbox{-}}\)stable. Then
\(T' \cap G\) is of the following form:

\begin{align*}
\begin{bmatrix}
t_1 &  &  &  &  &  \\
 & \ddots &  &  &  &  \\
 &  & t_n &  &  &  \\
 &  &  & t_n^{-1}&  &  \\
 &  &  &  & \ddots &  \\
 &  &  &  &  & t_1^{-1}
\end{bmatrix} 
\cong ({\mathbb{C}}^{\times}){ {}^{ \scriptscriptstyle\times^{n} }  }
.\end{align*}

Writing
\(G'/B' = \left\{{ F^\bullet \text{ complete flags}}\right\} = G' \cdot {\mathbb{C}}^\bullet\)
for the standard flag
\({\mathbb{C}}^\bullet \coloneqq(0 \subseteq {\mathbb{C}}^1 \subseteq {\mathbb{C}}^2 \subseteq \cdots \subseteq {\mathbb{C}}^{2n})\).
We can write this set as
\(\left\{{F^\bullet {~\mathrel{\Big\vert}~}(F^k)^\perp = F^{2n+1-k}}\right\}\),
where
\((F^k)^\perp \coloneqq\left\{{ x\in {\mathbb{C}}^{2n} {~\mathrel{\Big\vert}~}\omega(x, y) = 0 \,\,\forall y\in F^k}\right\}\).
Generally the former will be flags
\({\mathbb{C}}^{2n} =F^{2n} \to F^{2n-1} \to \cdots \to F^1\to 0\), and
this says we can describe this more compactly as flags
\({\mathbb{C}}^{2n} \to F^n \to F^{n-1} \to \cdots \to F^1\to 0\) where
the \(F^k\) are isotropic, by inserting their orthogonal complements
into the chain appropriately.

\end{remark}

\begin{question}

What are the Schubert varieties in \(G/B\)?

\end{question}

\begin{answer}

For \(w'\in W' = S_{2n}\), the Weyl group for
\(G' = \operatorname{GL}_{2n}\) and writing \(X' = G'/B'\), the Schubert
varieties are exactly \(X_{w'}' \cap G/B\). This is empty if there
exists a \(k\) with ???, and is \(X_W\) otherwise where
\(W \subseteq W'\) is
\(\left\{{ (w_1,\cdots, w_n) {~\mathrel{\Big\vert}~}w_1 + w_{2n} = 2n+1 }\right\}\).
For example, take \(\sigma = (1,3,2,4) \in W\), then
\(X_{W'}' = ({\mathbb{C}}^4 \to {\mathbb{C}}^3 \to F^2 \to {\mathbb{C}}^1)\)
and
\(X_W = ({\mathbb{C}}^4 \to {\mathbb{C}}^3 \to F_2 \to {\mathbb{C}}^1)\),
where \(F^2\) is a Lagrangian subspace of \({\mathbb{C}}^4\).

\end{answer}

\begin{remark}

This produces a large collection of normal varieties: start with flags
and add conditions.

\end{remark}

\hypertarget{wednesday-october-20}{%
\section{Wednesday, October 20}\label{wednesday-october-20}}

\begin{remark}

Last time: Schubert varieties for \(G \leq G'\) for
\(G \coloneqq{\mathsf{Sp}}_{2n}\) and
\(G' \coloneqq\operatorname{GL}_{2n}\). There are Weyl groups
\(W \leq W'\) where here \(W' = S_{2n}\) and
\(W = \left\{{w\in S_{2n} {~\mathrel{\Big\vert}~}w(k) + w(2n+1-k) = 2n+1-k }\right\}\).
For \({\mathsf{Sp}}_2 \leq \operatorname{GL}_4\), e.g.~we can take
\(w = (1,3 \mathrel{\Big|}2, 4)\) and
\(X_W = \left\{{F^\bullet \in G'/B' {~\mathrel{\Big\vert}~}{\mathbb{C}}^4\to {\mathbb{C}}^3 \to F^2\to {\mathbb{C}}^1 \to 0}\right\} \cong {\mathbb{P}}^1\).

\end{remark}

\begin{remark}

For \(G' = \operatorname{GL}_4\), we can produce a singular Schubert
variety. Take \(G/P\) for \(P = P_Y\) where
\(Y = \left\{{1, 3}\right\}\), so
\(G/P = {\operatorname{Gr}}_2({\mathbb{C}}^4)\). Take the following
Young diagram:

\includegraphics{figures/2021-10-20_14-03-50.png}

So
\(X_\lambda^Y = \left\{{E^2 \in {\operatorname{Gr}}_2({\mathbb{C}}^4) {~\mathrel{\Big\vert}~}\dim ({\mathbb{C}}^2 \cap E^2) \geq 1}\right\}\),
and \(X_W = \pi^{-1}(X_\lambda^Y)\). The minimal length permutation is
\(w' = (2,4 \mathrel{\Big|}1,3)\) (obtained from the Young diagram
above) and the maximal is \(w=(4,2 \mathrel{\Big|}3, 1)\). Note this
satisfies \(w(k) + 2(2n +1) = 2n + 1\) for \(n=2\) since
\(4+1 = 2+3 = 5\), so \(w\in W = W({\mathsf{Sp}}_4)\).

For this \(Y\), we have a map
\begin{align*}
\pi: G/B &\to G/P \\
F^\bullet &\mapsto F^2
,\end{align*}
where the full preimage is \(\pi^{-1}(P/P) = P/B\). Writing
\(X_W' = \left\{{F^\bullet \in G'/B' {~\mathrel{\Big\vert}~}\dim({\mathbb{C}}^2 \cap F^2) \geq 1 }\right\} \subseteq G'/B'\),
we can realize
\begin{align*}
X_W = \left\{{ F^\bullet \in G/B {~\mathrel{\Big\vert}~}\dim({\mathbb{C}}^2 \cap F^2) \geq 1, (F^1)^\perp = F^3, (F^2)^\perp = F^2}\right\}
.\end{align*}

\end{remark}

\begin{remark}

For \(G = {\mathsf{Sp}}_4\), \(S = \left\{{1, 2}\right\}\),
\(Y = \left\{{1}\right\}\), and
\(G/P_Y = \left\{{{\mathbb{C}}^4 \to F^2 \to 0 {~\mathrel{\Big\vert}~}F^2 = (F^2)^\perp}\right\}\)
since the \(1\in Y\) implies omitting \(F^1\), and we also omit
\((F^1)^\perp = F^3\). This yields the \textbf{Lagrangian flag variety}.

\end{remark}

\begin{remark}

Let \(s_1 = (2,1,4,3)\) and \(s_2 = (1,3,2,4)\), then
\(ws_1 = (4,2,3,1)(2,1,3,4) = (2,4,1,3)\) and notably
\(\ell(2,4,1,3) < \ell(4,2,3,1)\) and the length has strictly decreased.
So \(w\) is maximal length in \(wW_Y\).

We can conclude
\(X_w^Y = \left\{{F^2 \in {\mathcal{L}}\coloneqq{\operatorname{Gr}}^0_2({\mathbb{C}}^4) {~\mathrel{\Big\vert}~}\dim({\mathbb{C}}^2 \cap F^2) \geq 1 }\right\}\)
where \({\operatorname{Gr}}^0\) denotes isotropic subspaces. So this
yields a normal but not smooth variety.

\end{remark}

\hypertarget{statements-in-equivariant-mathsfkhbox-theory}{%
\subsection{\texorpdfstring{Statements in Equivariant
\({\mathsf{K}}{\hbox{-}}\)theory}{Statements in Equivariant \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory}}\label{statements-in-equivariant-mathsfkhbox-theory}}

\begin{quote}
See Chris-Ginzburg
\end{quote}

\begin{remark}

On flat pullback: for \(f:X\to Y\) a \(G{\hbox{-}}\)equivariant morphism
of \(G{\hbox{-}}\)spaces, if \(f\) is flat (so tensor-exact) then there
is a morphism of \(G{\hbox{-}}\)equivariant
\({\mathsf{K}}{\hbox{-}}\)theories:
\begin{align*}
f^*: K_i^G(Y) \to K_i^G(X)
\end{align*}
induced by an exact pullback functor
\begin{align*}
f^*: {\mathsf{Coh}}^G(Y) & \to {\mathsf{Coh}}^G(X) \\
{\mathcal{F}}&\mapsto f^* {\mathcal{F}}= {\mathcal{O}}_{X} \otimes_{f^{-1}{\mathcal{O}}_Y} f^{-1}{\mathcal{F}}
.\end{align*}

\end{remark}

\begin{slogan}

Flat implies sameness among fibers in a bundle.

\end{slogan}

\hypertarget{flat-pullback}{%
\subsection{Flat Pullback}\label{flat-pullback}}

\hypertarget{equivariant-descent}{%
\subsubsection{Equivariant Descent}\label{equivariant-descent}}

\begin{remark}

A principal \(G{\hbox{-}}\)bundle can mean several things. The
difference between local triviality in the Zariski vs étale topology
\footnote{Zariski locally trivial implies étale locally trivial.} Then
\(\pi:P\to X \in \mathop{\mathrm{Prin}}{ {\mathsf{Bun}}\qty{G} }\),
since étale implies flat there is an equivalence of categories
\({\mathsf{Coh}}(X) { { \, \xrightarrow{\sim}\, }}{\mathsf{Coh}}^G(P)\).
Thus there is an isomorphism
\(\pi^*: K(X) { { \, \xrightarrow{\sim}\, }}K^G(P)\).

\end{remark}

\hypertarget{restrictioninduction}{%
\subsubsection{Restriction/Induction}\label{restrictioninduction}}

\begin{remark}

For \(H \leq G\) a closed subgroup and \(X\) an \(H{\hbox{-}}\)space,
then \(G\overset{\scriptscriptstyle {H} }{\times} X\) is always an
algebraic variety. E.g. for \(X = {\operatorname{pt}}\),
\(G\overset{\scriptscriptstyle {H} }{\times} {\operatorname{pt}}= G/H\).
Note that there is a projection \(G\times X \to G\) where \(H\) acts
diagonally on the left and \(G\) is an \(H{\hbox{-}}\)space, and this
map is \(H{\hbox{-}}\) equivariant, so there is an induced map
\(G\overset{\scriptscriptstyle {H} }{\times}X\to G/H\). What's hard is
showing there are varieties. This is flat with fiber \(X\) since it's a
fiber bundle in our case.

For
\({\mathcal{F}}\in {\mathsf{Sh}}^G(G\overset{\scriptscriptstyle {H} }{\times} X)\)
a \(G{\hbox{-}}\)equivariant sheaf,

There is a functor
\begin{align*}
\operatorname{Ind}_H^G: {\mathsf{Coh}}^H(X) &\to {\mathsf{Coh}}^G(G\overset{\scriptscriptstyle {H} }{\times} X)
.\end{align*}

For \(p:G\times X\to X\) and \({\mathcal{F}}\in {\mathsf{Sh}}^H(X)\) an
\(H{\hbox{-}}\)equivariant sheaf, we can use a diagonal action to obtain
\(p^* {\mathcal{F}}\in {\mathsf{Sh}}^H(X)\) and write
\begin{align*}
\operatorname{Ind}_H^G = p^* {\mathcal{F}}\in {\mathsf{Coh}}(G\overset{\scriptscriptstyle {H} }{\times} X) { { \, \xrightarrow{\sim}\, }}{\mathsf{Coh}}^H(G\times H)
.\end{align*}
This defines a \(G{\hbox{-}}\)equivariant structure on
\(p^* {\mathcal{F}}\).

\end{remark}

\hypertarget{toward-the-demazure-character-formula-friday-october-22}{%
\section{Toward the Demazure Character Formula (Friday, October
22)}\label{toward-the-demazure-character-formula-friday-october-22}}

\begin{quote}
References: Chris-Ginzburg
\end{quote}

\begin{remark}

Recall that we discussed proper pushforward and flat pullback.

\end{remark}

\begin{remark}[on induction]

For \(H\leq G\in {\mathsf{Alg}}{\mathsf{Grp}}\) linear groups and
\(X\in {H}{\hbox{-}}{\mathsf{Spaces}}\), it is a fact that
\(G\overset{\scriptscriptstyle {H} }{\times} X \in {G}{\hbox{-}}{\mathsf{Spaces}}\).
There is a functor inducing an equivalence of categories:
\begin{align*}
\operatorname{Ind}_H^G: {\mathsf{Coh}}^H(X) { { \, \xrightarrow{\sim}\, }}{\mathsf{Coh}}^G(X)
,\end{align*}
yielding an isomorphism of groups
\(K_i^H(X) \to K_i^G(G\overset{\scriptscriptstyle {H} }{\times} X)\).
Induction can be constructed by quotienting the projection map:

\begin{center}
\begin{tikzcd}
    \textcolor{rgb,255:red,92;green,92;blue,214}{{\operatorname{pr}}^{-1}({\mathcal{F}})} &&& \textcolor{rgb,255:red,92;green,92;blue,214}{{\mathcal{F}}\in {\mathsf{Coh}}^H(X)} \\
    & {G\times X} && X \\
    \\
    \textcolor{rgb,255:red,92;green,92;blue,214}{\operatorname{Ind}_H^G({\mathcal{F}})\in{\mathsf{Coh}}^H(G\overset{\scriptscriptstyle {H} }{\times} X)} & {G\overset{\scriptscriptstyle {H} }{\times} X}
    \arrow["{\operatorname{pr}}", from=2-2, to=2-4]
    \arrow["{{-}/H}"', from=2-2, to=4-2]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-4, to=1-1]
    \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=4-1]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMSwxLCJHXFx0aW1lcyBYIl0sWzMsMSwiWCJdLFsxLDMsIkdcXG1peHtIfSBYIl0sWzMsMCwiXFxtY2YgXFxpbiBcXENvaF5IKFgpIixbMjQwLDYwLDYwLDFdXSxbMCwwLCJcXHByXFxpbnYoXFxtY2YpIixbMjQwLDYwLDYwLDFdXSxbMCwzLCJcXEluZF9IXkcoXFxtY2YpXFxpblxcQ29oXkgoR1xcbWl4e0h9IFgpIixbMjQwLDYwLDYwLDFdXSxbMCwxLCJcXHByIl0sWzAsMiwiXFx3YWl0L0giLDJdLFszLDQsIiIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFs0LDUsIiIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link
to Diagram}
\end{quote}

\end{remark}

\begin{remark}

There is also a restriction functor inducing
\(\mathop{\mathrm{Res}}: K_i^G(X) \to K_i^H(X)\):

\begin{center}
\begin{tikzcd}
    X && {G\overset{\scriptscriptstyle {H} }{\times} X} \\
    \\
    {H/H} && {G/H}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-1, to=3-1]
    \arrow[from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJYIl0sWzIsMCwiR1xcbWl4e0h9IFgiXSxbMCwyLCJIL0giXSxbMiwyLCJHL0giXSxbMCwxXSxbMCwyXSxbMiwzXSxbMSwzXV0=}{Link
to Diagram}
\end{quote}

Any linear \(G\in {\mathsf{Alg}}{\mathsf{Grp}}\) can be written as
\(G\cong R\rtimes U\) where \(R\) is reductive and \(U\) is unipotent.

\end{remark}

\begin{proposition}[?]

For any \(X\in {G}{\hbox{-}}{\mathsf{Spaces}}\),
\begin{align*}
K^G(X) \cong K^R(X)
.\end{align*}

\end{proposition}

\begin{slogan}

Only the reductive groups matter for equivariant
\({\mathsf{K}}{\hbox{-}}\)theory.

\end{slogan}

\begin{proof}[?]

Define the morphism \(K^G(X)\to K^R(X)\) by forgetting the action away
from the subgroup \(R \leq X\):
\begin{align*}
G\overset{\scriptscriptstyle {R} }{\times} X &\xrightarrow{\phi} G/R \times X \\
[g, x] &\mapsto (gR/R, gx)
.\end{align*}
This induces
\begin{align*}
{\mathsf{K}}^R(X) \xrightarrow{\operatorname{Ind}_R^G} {\mathsf{K}}^G(G\overset{\scriptscriptstyle {R} }{\times} X) \underset{{\mathsf{K}}\phi }{{ { \, \xrightarrow{\sim}\, }}} {\mathsf{K}}^G(G/R \times X) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}^G(X)
,\end{align*}
using that \(G/R\) is affine.

\begin{quote}
More generally, for
\(E\to X\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }^G\), the fibers
are contractible and thus \({\mathsf{K}}^G(E) \cong {\mathsf{K}}^G(X)\).
See the Thom isomorphism, referenced in Borbo-Brylinksi-MacPherson.
\end{quote}

\end{proof}

\begin{remark}

Let \(\pi: G/B \to G/P\) where \(P\) corresponds to the simple
reflection \(s\), so \(P\) is the smallest parabolic not equal to the
Borel. Then

\begin{itemize}
\tightlist
\item
  Any map between projective varieties is proper, so \(\pi\) is proper
  and the fibers are copies of \({\mathbb{P}}^1\),
  i.e.~\(\pi^{-1}(gP/P) = gP/B \cong {\mathbb{P}}^1\).
\item
  \(\pi\) is smooth in the sense of Hartshorne, i.e.~so smooth fibers
  that are ``the same''.
\end{itemize}

Consequently, \(\pi\) is flat, and
\(G/B \cong G\overset{\scriptscriptstyle {P} }{\times} P/B \to G/P\)
with \(G/B\to G/P\) flat. We can push forward along proper maps and pull
back along flat maps, so here we can do both. So define a map
\begin{align*}
D: {\mathsf{K}}^G(G/B) &\to {\mathsf{K}}^G(G/B) \\
[{\mathcal{F}}] &\mapsto \pi^*\pi_*[{\mathcal{F}}]
.\end{align*}
Note that this factors as
\({\mathsf{K}}^G(G/B) \xrightarrow{\pi_*} {\mathsf{K}}^G(G/P) \xrightarrow{\pi_*} {\mathsf{K}}^G(G/B)\).
The question is now what \(\pi^* \pi_* [{\mathcal{F}}]\) actually is.

\end{remark}

\begin{slogan}

The idea: we can recover representations as
\({\mathsf{K}}^G({\operatorname{pt}})\), which is hard, so we apply
these \(D\) operators to larger parabolics to get to a point one step at
a time.

\end{slogan}

\begin{remark}

We have
\(A(T) = {\mathbb{Z}}[X(T)] \cong {\mathsf{K}}^T({\operatorname{pt}})\)
for \(A(T)\) representations of the torus. On notation: write
\(\lambda \in X(T)\) as \(e^{\lambda} \in A(T)\). Note that
\({\mathsf{K}}^P(P/B) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}^P(P\overset{\scriptscriptstyle {B} }{\times} {\operatorname{pt}}) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}^B({\operatorname{pt}})\)
and
\(A(T) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}^T({\operatorname{pt}})\),
so writing \(B = T\rtimes U\), there is an isomorphism
\begin{align*}
A(T) &{ { \, \xrightarrow{\sim}\, }}{\mathsf{K}}^P(P/B) \\
C^{\lambda} &\mapsto [P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}] \mapsto [G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}]
,\end{align*}
which is a composition
\(\operatorname{Ind}_P^B \circ \operatorname{Ind}_T^P\). One can regard
this as a line bundle on \({\mathbb{CP}}^1\) via the projection
\(P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda} \to P/B { { \, \xrightarrow{\sim}\, }}{\mathbb{P}}^1\).

\end{remark}

\begin{remark}

A trick: recovering \({\mathsf{K}}^G\) from \({\mathsf{K}}^T\) and the
Weyl group action on it. This is why we reduce to \({\mathsf{K}}^T\) so
often! Write \({\mathsf{K}}^G({\operatorname{pt}}) = R(G)\) on one hand
and \(A(T)^W\) on the other (taking Weyl group invariants), define a map
\([V] \mapsto \sum_{\lambda \in X(T)} n_\lambda e^{\lambda}\). Now
assemble some maps:

\begin{center}
\begin{tikzcd}
    {{\mathsf{K}}^L({\operatorname{pt}})} && {{\mathsf{K}}^P({\operatorname{pt}})} && {{\mathsf{K}}^G(G/P)} \\
    \\
    {A(T)} && {{\mathsf{K}}^P(P/B)} && {{\mathsf{K}}^G(G/B)}
    \arrow["\cong", from=1-1, to=1-3]
    \arrow["\cong", from=1-3, to=1-5]
    \arrow["{\pi^*}", from=3-3, to=1-3]
    \arrow["{D_s}", from=3-1, to=1-1]
    \arrow["{\pi_*}"', from=3-5, to=1-5]
    \arrow["\cong", from=3-1, to=3-3]
    \arrow["\cong", from=3-3, to=3-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNixbMCwwLCJcXEteTChcXHB0KSJdLFsyLDAsIlxcS15QKFxccHQpIl0sWzQsMCwiXFxLXkcoRy9QKSJdLFswLDIsIkEoVCkiXSxbMiwyLCJcXEteUChQL0IpIl0sWzQsMiwiXFxLXkcoRy9CKSJdLFswLDEsIlxcY29uZyJdLFsxLDIsIlxcY29uZyJdLFs0LDEsIlxccGleKiJdLFszLDAsIkRfcyJdLFs1LDIsIlxccGlfKiIsMl0sWzMsNCwiXFxjb25nIl0sWzQsNSwiXFxjb25nIl1d}{Link
to Diagram}
\end{quote}

What is \(D_s(e^{\lambda})\)? By defining of pushforward along proper
morphisms, we can write Using these identifications, write
\begin{align*}
\pi_*[G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}] 
&= \pi_* [P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}] \\
&= \sum_i (-1)^i [{\mathbb{R}}^i \pi_* (P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}) ] \\
&= [H^0(P/B, e^{\lambda})] - [H^1(P/B, e^{ \lambda})] \\
&= [H^0({\mathbb{P}}^1, e^{\lambda})] - [H^1({\mathbb{P}}^1, e^{ \lambda}) ]
.\end{align*}
Recall that for \({\mathcal{O}}(k)\), we have a pairing
\(-1, 0 \iff -2, 1 \iff -3, \cdots\) and
\({\left\langle { \lambda},~{ \alpha {}^{ \vee }} \right\rangle} = k\).

\end{remark}

\begin{remark}

Next time: the Demazure character formula.

\end{remark}

\hypertarget{demazure-character-formula-monday-october-25}{%
\section{Demazure Character Formula (Monday, October
25)}\label{demazure-character-formula-monday-october-25}}

\begin{quote}
See Anderson 1985
\end{quote}

\begin{remark}

Today:
\(A(T) = {\mathbb{Z}}[X(T)] \cong {\mathsf{K}}_T({\operatorname{pt}})\),
where we write characters multiplicatively as \(e^{\lambda}\). For
\(\pi: G/B\to G/P\) for \(P\) a simple parabolic corresponding to
\(s\in S\), we can push-pull to get an endomorphism of
\({\mathsf{K}}_G(G/B)\), using that this morphism is both flat and
proper. The goal is to compute
\(\pi^* \pi_*[G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}]\),
and the major tool in \({\mathsf{K}}{\hbox{-}}\)theory is induction.
Write
\(G/B = G\overset{\scriptscriptstyle {P} }{\times} P/B = G\overset{\scriptscriptstyle {B} }{\times} {\operatorname{pt}}\)
and \(P = LU\), then there is a diagram

\begin{center}
\begin{tikzcd}
    {[G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}]} && {[P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}]} \\
    {{\mathsf{K}}_G(G/B)} && {{\mathsf{K}}_L(P/B)} && {{\mathsf{K}}_T({\operatorname{pt}})} & {e^{\lambda}} \\
    \\
    {{\mathsf{K}}_G(G/P)} && {{\mathsf{K}}_L(P/P)} && {{\mathsf{K}}_T({\operatorname{pt}})^{W_S} \subseteq {\mathsf{K}}_T({\operatorname{pt}})} & {?}
    \arrow[from=2-1, to=2-3]
    \arrow[from=2-3, to=2-5]
    \arrow[from=4-1, to=4-3]
    \arrow[from=4-3, to=4-5]
    \arrow[from=2-5, to=4-5]
    \arrow[from=2-3, to=4-3]
    \arrow["{\pi_*}"{description}, from=2-1, to=4-1]
    \arrow[maps to, from=2-6, to=4-6]
\end{tikzcd}
\end{center}

\end{remark}

\begin{fact}

\({\mathsf{K}}_G({\operatorname{pt}}) = {\mathsf{K}}_T({\operatorname{pt}})^W\).

\end{fact}

\begin{remark}

Writing \(W = X^T \subseteq X = G/B\), we can use something due Bill
Graham. It's a fact that
\(i^* {\mathsf{K}}_T(X) \to {\mathsf{K}}_T(X^T)\) is injective, and Bill
shows \(i^*\) is an isomorphism after inverting certain elements.

\end{remark}

\begin{corollary}[Chris-Ginzburg, 5.11.3]

The composite \(i^*i_* {\mathsf{K}}_T(X^T) \to {\mathsf{K}}_T(X^T)\) is
multiplication by \(\lambda_T\), so here \(\lambda_{-1}\). Moreover
\begin{align*}
\lambda_T = \sum (-1)^i \Lambda^i N {}^{ \vee }\in {\mathsf{K}}_T(X_T)
\end{align*}
where \(N {}^{ \vee }\) is the conormal.

\end{corollary}

\begin{example}[?]

For \(X = {\mathbb{P}}^1\) and \(W = \left\{{1, s}\right\}\), we have

\begin{itemize}
\tightlist
\item
  \({\mathbf{T}}_{B/B}(G/B) = {\mathfrak{g}}/{\mathfrak{b}}= {\mathbb{C}}_{-\alpha}\),
\item
  \({\mathbf{T}}_{sB/B}(G/B) = {\mathfrak{g}}/s{\mathfrak{b}}= {\mathbb{C}}_{\alpha}\),
\item
  \(N_1 {}^{ \vee }= {\mathbb{C}}_{\alpha}\),
\item
  \(N_s {}^{ \vee }= {\mathbb{C}}_{-\alpha}\).
\end{itemize}

\end{example}

\begin{proposition}[?]

A formula due to Bill, there is an element:
\begin{align*}
{\mathsf{K}}_T(X) \ni \alpha = \sum_{w\in X^T} (i_w)_* \qty{ (i_w)^* \alpha \over \lambda_{-1} (N {}^{ \vee }_w)}
.\end{align*}

\end{proposition}

\begin{remark}

Write \(\pi:G/B\to G/P\) and its restriction \(P/B\to P/P\). Pullbacks
are easy enough to compute, and we have formulas

\begin{itemize}
\tightlist
\item
  \((i_1)^* [P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}] = [B\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}]\),
\item
  \((i_s)^*[P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}] = [sB \overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}]\).
\end{itemize}

For
\([P\overset{\scriptscriptstyle {B} }{\times} \in {\mathbb{C}}_{\lambda}]\),
we can compute
\begin{align*}
\pi_*[ P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}] &=
\pi_* \sum_{1, s} (i_w)_* \qty{ (i_w)^* [P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}] \over \lambda_{-1} (N_w {}^{ \vee }) } \\
&=
\pi_* \qty{
(i_1)_* \qty{ [B\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}] \over 1 - e^{\alpha}}
+
(i_s)_* \qty{ [sB\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}] \over 1 - e^{-\alpha}}
}\\
&=
\pi_* \qty{
(i_1)_* \qty{ e^{ \lambda} \over 1- e^{ \lambda}}
+
(i_s)_* \qty{ e^{ s\lambda} \over 1- e^{ \lambda}}
}\\
&= 
\qty{ e^{ \lambda} \over 1- e^{ \lambda}}
+
\qty{ e^{ s\lambda} \over 1- e^{ \lambda}}
\in A(T)
.\end{align*}

\end{remark}

\begin{proposition}[?]

\begin{align*}
\pi_* [P\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda}] = {e^{\lambda} - s^{s \lambda + \alpha} \over 1 - e^{ \alpha}}
.\end{align*}

\end{proposition}

\begin{proof}[?]

Let \(q = e^{\alpha}\) and
\(k \coloneqq{\left\langle {\lambda},~{ \alpha {}^{ \vee }} \right\rangle}\).
Note that
\begin{align*}
s \lambda- \lambda = \lambda- {\left\langle {\lambda},~{ \alpha {}^{ \vee }} \right\rangle} - \lambda = -{\left\langle {\lambda},~{\alpha {}^{ \vee }} \right\rangle} \alpha
.\end{align*}
Then
\begin{align*}
(e^{ \lambda} - e^{ s \lambda+ \alpha})( 1- e^{ \alpha}) 
= e^{\lambda}(1 - e^{ - \alpha}) + e^{s \lambda}(1 - e^{ \alpha})
,\end{align*}
and we can write the RHS as
\begin{align*}
e^{ \lambda}\qty{ 1 - e^{ s \lambda- \lambda+ \alpha} \over 1 - e^{- \alpha }} 
=
e^{ \lambda}\qty{ 1 - q^{1-k} \over 1 - q} 
= e^{\lambda} c(q)
\end{align*}
where
\begin{align*}
c(q) 
&= 
\begin{cases}
1 + q + \cdots + q^{-k} & k\leq 0 \\
0 & k=0 \\
-\qty{q^{1-k} + q^{2-k} + \cdots + q^{-1}}& k\geq 1.
\end{cases}\\ \\
&=
\begin{cases}
e^{ \lambda} + e^{ \lambda+ \alpha} + \cdots + e^{s \lambda}& k\leq 0 \\
0 & k=0 \\
-\qty{ e^{s \lambda+ \alpha} + e^{s \lambda+ 2 \alpha} + \cdots + e^{s \lambda + (k-1) \alpha} }& k\geq 1.
\end{cases}
.\end{align*}

\end{proof}

\begin{remark}

By Kumar,
\begin{align*}
D_s(e^{ \lambda}) \coloneqq{e ^{ \lambda} - e^{s \lambda- \alpha} \over 1 - e^{- \alpha}}
,\end{align*}
where \(e^{ \lambda}\) corresponds to \({\mathcal{L}}( \lambda)\).

\end{remark}

\begin{theorem}[8.?]

For any \(w\in W\), not necessarily reduced, and finite dimensional
\(M\) of \(B\),

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  There is an Euler characteristic formula
  \begin{align*}
  \chi(Z_w, {\mathcal{L}}_w(M)) = \mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_w(\mkern 1.5mu\overline{\mkern-1.5mu\operatorname{ch}M\mkern-1.5mu}\mkern 1.5mu)
  ,\end{align*}
  where \(\chi\) is given by
  \(\sum (-1)^p \operatorname{ch}\qty{H^p(Z_w, {\mathcal{L}}_w(M))} \in A(T)\).
\item
  \(\chi(X_w, {\mathcal{L}}_w( \lambda)) = \mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_w(e^{\lambda})\).
\end{enumerate}

Then if \(\lambda \in D_{\mathbb{Z}}\),

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{2}
\tightlist
\item
  \(\operatorname{ch}H^0(X_w, {\mathcal{L}}_w( \lambda)) = \mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_w (e ^{\lambda})\)
\item
  \(\operatorname{ch}L_w^{\max}( \lambda) = D_w(e^{ \lambda})\).
\end{enumerate}

\end{theorem}

\hypertarget{wednesday-october-27}{%
\section{Wednesday, October 27}\label{wednesday-october-27}}

\begin{remark}

If \(H \leq G \in {\mathsf{Alg}}{\mathsf{Grp}}\) is a closed linear
subgroup and \(Y\in{G}{\hbox{-}}{\mathsf{Spaces}}\), then there is a
commuting diagram

\begin{center}
\begin{tikzcd}
    {G\overset{\scriptscriptstyle {H} }{\times}Y} && {G/H\times Y} \\
    \\
    & {G/H}
    \arrow["{{\operatorname{pr}}_1}", from=1-3, to=3-2]
    \arrow["\pi"', from=1-1, to=3-2]
    \arrow["\cong", tail reversed, from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJHXFxtaXh7SH1ZIl0sWzIsMCwiRy9IXFx0aW1lcyBZIl0sWzEsMiwiRy9IIl0sWzEsMiwiXFxwcl8xIl0sWzAsMiwiXFxwaSIsMl0sWzAsMSwiXFxjb25nIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiYXJyb3doZWFkIn19fV1d}{Link
to Diagram}
\end{quote}

The isomorphism \(\phi\) is given by
\begin{align*}
[g ,y] &\mapsto (\mkern 1.5mu\overline{\mkern-1.5mug\mkern-1.5mu}\mkern 1.5mu, gy) \\
[g, g^{-1}y] &\mapsfrom (g, y)
.\end{align*}

More generally, if \(Y \subseteq X\) one often has \(H \leq G\) with
\(H\curvearrowright Y\) and \(G\curvearrowright X\). In this case,
\(\phi: G\overset{\scriptscriptstyle {H} }{\times} Y \xrightarrow{\phi} G/H \times X\)
may be an embedding instead.

\end{remark}

\begin{proposition}[?]

For \(G\) connected reductive and \(T\leq G\) is a maximal torus,
\begin{align*}
R_G = 
{\mathsf{K}}_G({\operatorname{pt}}) \cong {\mathsf{K}}_T({\operatorname{pt}})^W
= R_T^W
.\end{align*}

\end{proposition}

\begin{slogan}

To compute \(G{\hbox{-}}\)equivariant \({\mathsf{K}}{\hbox{-}}\)theory,
it suffices to understand \(T{\hbox{-}}\)equivariant
\({\mathsf{K}}{\hbox{-}}\)theory and the action of the Weyl group.

\end{slogan}

\begin{proof}[?]

Define \(\rho R_G\to R_T\) by restriction to \(T\), so explicitly
\(\rho[v] = \sum_{ \lambda} m_{\lambda}e^{\lambda} \in R_T^W\) where the
\(m_{\lambda}\) are the multiplicities of \(e^{\lambda}\) in
\(V_{ \lambda}\). Set \(G^{sr}\) to be the \textbf{semisimple regular}
elements in \(G\). Note that a regular element \(t\in T\) satisfies
\(t\not\in \ker \alpha\), and

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(G^{sr} \hookrightarrow G\) is open and dense.
\item
  Every \(g\in G^{rs}\) is conjugate to some \(t\in T\).
\end{enumerate}

Let \(f\in {\mathbb{C}}[G]^G\) be function invariant under
\(G{\hbox{-}}\)conjugation, i.e.~a class function, and suppose
\({ \left.{{f}} \right|_{{T}} } = 0\). By (ii),
\({ \left.{{f}} \right|_{{G^{sr}}} } = 0\), so by (i) \(f\equiv 0\) on
\(G\) since \(f\) is continuous and zero on a dense subset. There is a
diagram:

\begin{center}
\begin{tikzcd}
    {R(G)} && {R(T)^W} \\
    \\
    {{\mathbb{C}}[G]^G} && {{\mathbb{C}}[T]^W}
    \arrow["\rho", from=1-1, to=1-3]
    \arrow["{{-}\otimes_{\mathbb{Z}}{\mathbb{C}}}"', from=1-1, to=3-1]
    \arrow["{\therefore }"', hook, from=3-1, to=3-3]
    \arrow["{{-}\otimes_{\mathbb{Z}}{\mathbb{C}}}", from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJSKEcpIl0sWzIsMCwiUihUKV5XIl0sWzAsMiwiXFxDQ1tHXV5HIl0sWzIsMiwiXFxDQ1tUXV5XIl0sWzAsMSwiXFxyaG8iXSxbMCwyLCJcXHdhaXRcXHRlbnNvcl9cXFpaIFxcQ0MiLDJdLFsyLDMsIlxcdGhlcmVmb3JlICIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEsMywiXFx3YWl0XFx0ZW5zb3JfXFxaWiBcXENDIl1d}{Link
to Diagram}
\end{quote}

Here the bottom map is injective by the previous argument. To prove
\(\rho\) is surjective, fix \(f\in R(T)^W\), then we'll produce an
\(h\in {\mathbb{C}}[G]^G\) such that
\({ \left.{{h}} \right|_{{T}} } = f\). Choosing \(B \supseteq T\) a
Borel, then for any such Borel containing \(T\) is a canonical
isomorphism \(T \subseteq B \to B/U\) where we write \(B = T\rtimes U\).
So identify \(f\) with an element of \(R(B/U)^W\). Let
\(Z\coloneqq G\overset{\scriptscriptstyle {B} }{\times} B\), and instead
of having the same action of \(B\) on both factors (which would be
isomorphic to \(G\) by mapping to \(B/B\) with fiber \(G\)) let
\(B\curvearrowright G\) by conjugation. Define a map
\begin{align*}
\mu: Z\to G \\
[g, b] &\mapsto gbg^{-1}
,\end{align*}
which is a \(G{\hbox{-}}\)equivariant algebraic morphism. Then
\(\mu^{-1}(g) = \left\{{B' \supseteq g}\right\}\) are the Borels
containing \(g\): note the similarity to the Springer resolution with
the nilpotent radical.

\begin{exercise}[?]

Prove this -- a hint is that
\(G\overset{\scriptscriptstyle {B} }{\times} B \xrightarrow{\subseteq} G/B\times G\).

\end{exercise}

Note the two extremal cases:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \(\mu^{-1}(1) = G/B\).
\item
  For \(g\in G^{sr}\) regular semisimple, use conditions on dimensions
  of centralizers and \(\dim T \coloneqq\dim Z(T)\), how many Borels
  contain a fixed maximal torus \(T\)? There are at least two, since
  \(T \subseteq B \implies T \subseteq B^-\). One can think of the flag
  variety as parameterizing Borels, so these correspond to
  \(T{\hbox{-}}\)fixed points in the flag variety. The key is that \(W\)
  acts simply transitively, so \(\mu^{-1}(g) \cong W\).
\end{enumerate}

Define a map
\begin{align*}
\nu: Z = G\overset{\scriptscriptstyle {B} }{\times} B &\to (G\times U)\overset{\scriptscriptstyle {B} }{\times} B 
\xrightarrow{{-}/U} G\overset{\scriptscriptstyle {B} }{\times} B/U 
{ { \, \xrightarrow{\sim}\, }}G/B \times B/U 
\xrightarrow{{\operatorname{pr}}_2} B/U \\
[g,b] &\mapsto (\mkern 1.5mu\overline{\mkern-1.5mug\mkern-1.5mu}\mkern 1.5mu, \mkern 1.5mu\overline{\mkern-1.5mugb\mkern-1.5mu}\mkern 1.5mu) \xrightarrow{\text{trivial action}} \mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu
,\end{align*}
where we've used that relevant actions commute. Note that this composite
map is rare, but allows defining an \textbf{abstract Cartan}. We can
then pull back \(f\) to a regular function on \(Z\), so set
\(\tilde f \coloneqq\nu_* f\), so
\(\tilde f[g, b] = f(\mkern 1.5mu\overline{\mkern-1.5mub\mkern-1.5mu}\mkern 1.5mu)\).

\begin{claim}

\(\tilde f\in {\mathbb{C}}[Z]^B\).

\end{claim}

Next restrict \(\tilde f\) to \(Z^{sr} = \mu^{-1}(G^{sr})\), then
\(W\curvearrowright Z^{sr}\) freely and \(\nu\) is
\(W{\hbox{-}}\)equivariant. Since \(f\) is \(W{\hbox{-}}\)invariant,
\({ \left.{{ \tilde f}} \right|_{{Z^{sr}}} }\) to be
\(W{\hbox{-}}\)invariant and
\({ \left.{{ \tilde f}} \right|_{{Z^{sr}}} } \in {\mathbb{C}}[Z^{sr}]^W\).

\begin{fact}

If \(\xi:X\to Y\) is a quotient by a free action of a finite group, then
\(\xi\) is \textbf{generically Galois},
i.e.~\(\mu^*: {\mathbb{C}}(G^{sr}) { { \, \xrightarrow{\sim}\, }}{\mathbb{C}}(Z^{sr})^W\).

\end{fact}

\begin{claim}

\(h\) is regular on \(G\), i.e.~\(h\in {\mathbb{C}}[G]\).

\end{claim}

\begin{quote}
See Chriss-Ginzburg 3.1.3.
\end{quote}

\end{proof}

\begin{remark}

Next time: equivariant cohomology.

\end{remark}

\hypertarget{equivariant-mathsfkhbox-theory-of-gp-monday-november-01}{%
\section{\texorpdfstring{Equivariant \({\mathsf{K}}{\hbox{-}}\)theory of
\(G/P\) (Monday, November
01)}{Equivariant \{\textbackslash mathsf\{K\}\}\{\textbackslash hbox\{-\}\}theory of G/P (Monday, November 01)}}\label{equivariant-mathsfkhbox-theory-of-gp-monday-november-01}}

\begin{remark}

We'll stick to the finite-type case for today. Setup: let
\(G\in {\mathsf{Alg}}{\mathsf{Grp}}_{/ {{\mathbb{C}}}}\) be connected,
semisimple, simply connected, with \(T\leq G\) a maximal torus. Goal:
describe \({\mathsf{K}}_T(G/P)\).

\end{remark}

\begin{remark}

Note that
\begin{align*}
{\mathsf{K}}_G(G/B) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_G(G\overset{\scriptscriptstyle {B} }{\times} {\operatorname{pt}}) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_B({\operatorname{pt}}) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_T({\operatorname{pt}})
,\end{align*}
which we sometimes write as \({\mathsf{K}}_T\) or \(A(T)\), the
representation ring of \(T\). General pattern: for
\({\mathsf{K}}_G({-})\), look at \({\mathsf{K}}_T({-})^W\) instead,
using that
\({\mathsf{K}}_G({\operatorname{pt}}) = {\mathsf{K}}_T({\operatorname{pt}})^W = A(T)^W\).
Writing \(P = LU\supseteq T\) for \(P\) a parabolic and \(L\) a Levi, we
have
\begin{align*}
{\mathsf{K}}_G(G/P) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_P({\operatorname{pt}}) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_L({\operatorname{pt}}) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_T({\operatorname{pt}})^{W_Y} { { \, \xrightarrow{\sim}\, }}A(T)^{W_Y}
.\end{align*}

Thus there is a chain of isomorphisms:
\begin{align*}
{\mathsf{K}}_T(G/P) 
&{ { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_B(G/P) && \text{doesn't see unipotent radical} \\
&{ { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_G(G\overset{\scriptscriptstyle {B} }{\times} G/P) && \text{induction}\\
&{ { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_G(G/B \overset{\scriptscriptstyle {B} }{\times} G/P) && \text{trivialization for algebraic fiber bundles} \\
&{ { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_G(G/B) \otimes_{{\mathsf{K}}_G({\operatorname{pt}})} {\mathsf{K}}_G(G/P) && \text{Kunneth} \\
&{ { \, \xrightarrow{\sim}\, }}A(T) \otimes_{A(T)^W} A(T)^{W_Y}
.\end{align*}
Note that
\(A(T) = {\mathbb{Z}}[X(T)] = {\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{\ell} } }\)
for some \(\ell\).

\end{remark}

\begin{remark}

This formula may hold in more generality, but we're sticking with what's
in the literature for now.

\end{remark}

\begin{remark}

Phrasing this in terms of equivariant line bundles: starting with
\(\lambda\in X(T)\), we write it as \(e ^{\lambda} \in A(T)\), and we
have two morphisms \(A(T) \to {\mathsf{K}}_T(G/B)\):

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(F_1: e^{ \lambda} \to G/B \times {\mathbb{C}}_{\lambda} \in {\mathsf{K}}_T(G/B)\).
\item
  \(F_2: e^{\lambda} \to G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{ \lambda}\in {\mathsf{K}}_T(G/B)\).
\end{enumerate}

Note that the latter can be projected onto \(G/B\). If
\(e^{\lambda}\in R(G) = A(T)^W\), then
\(G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{\lambda} \cong G/B \times {\mathbb{C}}_{\lambda}\)
since the \(B{\hbox{-}}\)action extends to a \(G{\hbox{-}}\)action. So
these assemble to a map
\begin{align*}
F_1 \otimes F_2: A(T) \otimes_{A(T)^W} A(T) \to {\mathsf{K}}_T(G/B)
.\end{align*}
The claim is that this is equivalent to the isomorphism from above.

\end{remark}

\hypertarget{equivariant-cohomology}{%
\subsection{Equivariant Cohomology}\label{equivariant-cohomology}}

\begin{quote}
Perhaps don't try to learn this from Kumar as a first pass! See
\href{https://people.math.osu.edu/anderson.2804/eilenberg/}{Anderson-Fulton}
for a good treatment. For fiber bundles, see \emph{Husemoller}. For
algebraic topology, see May's ``Concise Course..'', chapter 18.
\end{quote}

\begin{slogan}

Studying the equivariant geometry of a space \(X\) is the same as
studying fiber bundles with fiber \(X\).

\end{slogan}

\begin{remark}

Recalling some notions of axiomatic cellular cohomology: fix
\(M\in {\mathsf{Ab}}{\mathsf{Grp}}\) and consider pairs
\((X, A)\in {\mathsf{Top}}\). Then there exist functors
\(H^k(X, A; M): {\mathsf{ho}}{\mathsf{Top}}{ {}^{ \scriptscriptstyle\times^{2} } } \to {\mathsf{Ab}}{\mathsf{Grp}}\)
with natural transformations \(\delta: H^k(A; M) \to H^{k+1}(X, A; M)\),
where \(H^k(A; M)\coloneqq H^k(A, \emptyset; M)\). These satisfy and are
characterized by a set of 5 axioms, which we'll omit. Note that these
constructions will work for any space we run into in this setting.

\end{remark}

\begin{exercise}[?]

If \(B \subseteq A \subseteq X\), show that there is a LES

\begin{center}
\begin{tikzcd}
    {H^{k+1}(X, A; M)} && \cdots \\
    \\
    {H^k(X, A; M)} && {H^k(X, B; M)} && {H^k(A, B; M)}
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=1-1]
    \arrow[from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwyLCJIXmsoWCwgQTsgTSkiXSxbMiwyLCJIXmsoWCwgQjsgTSkiXSxbNCwyLCJIXmsoQSwgQjsgTSkiXSxbMCwwLCJIXntrKzF9KFgsIEE7IE0pIl0sWzIsMCwiXFxjZG90cyJdLFswLDFdLFsxLDJdLFsyLDNdLFszLDRdXQ==}{Link
to Diagram}
\end{quote}

\end{exercise}

\begin{definition}[Equivariant cohomology]

Let \(G\in \operatorname{Lie}{\mathsf{Grp}}\) with
\(G\curvearrowright X\in {\mathsf{Top}}\) acting on the left.\footnote{Note
  that
  \({\mathsf{Alg}}{\mathsf{Grp}}_{/ {{\mathbb{C}}}} \leq \operatorname{Lie}{\mathsf{Grp}}\)!}
Write \(E\in {\mathsf{Top}}\) for any contractible space with a free
right \(G{\hbox{-}}\)action, then define the
\textbf{\(G{\hbox{-}}\)equivariant cohomology} of \(X\) as
\begin{align*}
H_G^k(X) \coloneqq H^k(E \overset{\scriptscriptstyle {G} }{\times} X)
.\end{align*}

\end{definition}

\begin{fact}

Some facts:

\begin{itemize}
\tightlist
\item
  \(X \simeq E\times X\).
\item
  \(H_G^k(X)\) does not depend on the homotopy representative of \(E\)
\item
  \({\mathbf{B}}G \coloneqq E/G\) is the \textbf{classifying space} of
  \(G\).
\item
  If \(\xi:X\to Y\) equivariant with respect to \(\phi:G\to H\), there
  is a map
  \(EG\overset{\scriptscriptstyle {G} }{\times} X\to EH\overset{\scriptscriptstyle {H} }{\times} Y\)
  which induces \(\xi^*: H_H^*(Y) \to H_G^*(X)\). In particular,
  \(X\to {\operatorname{pt}}\) always exists, which is why
  \(H^*({\mathbf{B}}G)\) plays a large role.
\item
  If \(G\supseteq H\) and \(EG\) is given, then \(EH = EG\).
\end{itemize}

\end{fact}

\begin{example}[?]

\begin{align*}
H_G^*({\operatorname{pt}}) { { \, \xrightarrow{\sim}\, }}H^*(E\overset{\scriptscriptstyle {G} }{\times} {\operatorname{pt}}) { { \, \xrightarrow{\sim}\, }}H^*(E/G) { { \, \xrightarrow{\sim}\, }}H^*({\mathbf{B}}G)
.\end{align*}

\end{example}

\begin{example}[?]

Examples of \({{\mathbf{B}}G}\):

\begin{itemize}
\tightlist
\item
  For \(G = {\mathbb{C}}^n\), we have
  \({\mathsf{E} G}= G = {\mathbb{C}}^n\) and
  \(H_G^*({\operatorname{pt}}) = H^*({\operatorname{pt}})= M\).
\item
  For \(G = {\mathbb{C}}^{\times}\),
  \(E = {\mathbb{C}}^\infty\setminus\left\{{0}\right\}\) which is a
  contractible Ind-variety, and
  \({{\mathbf{B}}G}= {\mathsf{E} G}/G = {\mathbb{P}}^\infty_{/ {{\mathbb{C}}}}\).
\end{itemize}

\end{example}

\hypertarget{chern-classes-and-intersection-theory-wednesday-november-03}{%
\section{Chern Classes and Intersection Theory (Wednesday, November
03)}\label{chern-classes-and-intersection-theory-wednesday-november-03}}

\hypertarget{chern-classes}{%
\subsection{Chern Classes}\label{chern-classes}}

\begin{quote}
See Eisenbud and Harris, 3264 and All That, and Fulton.
\end{quote}

\begin{theorem}[Klein's Transversality Theorem]

Let \(G \in {\mathsf{Alg}}{\mathsf{Grp}}\) act transitively on \(X\)
over
\(k = { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }\)
with \(\operatorname{ch}k = 0\) and let \(Y \leq X\) be a subvariety.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(Z\leq X\) is a subvariety then there is an open dense subset of
  group elements \(U \subseteq G\) such that \(gZ \pitchfork Y\)
  generically.
\item
  If \(\phi: Z\to X\) is a morphism of varieties, then for a generic
  \(g\in G\), the preimage \(\phi^{-1}(gY)\) is generally reduced and is
  of the same codimension as \(Y\).
\item
  If \(G\) is affine then \([gY] = [Y]\) in the Chow group \(A(X)\).
\end{enumerate}

\end{theorem}

\begin{remark}

See ELC article, a consequence is Bertini's theorem.

\end{remark}

\begin{lemma}[?]

Suppose
\({\mathcal{E}}\in { {\mathsf{Bun}}\qty{\operatorname{GL}_{r}} }_{/ {Y}}\)
and let \(1\leq i\leq r\). Let \(\sigma_0, \cdots, \sigma_{r-i}\) be
global sections of \({\mathcal{E}}\) and
\(Y_{\sigma} = Y( \sigma_0 \vee\cdots \vee\sigma_{r-i})\) be the
degeneracy locus where they are linearly dependent, so
\(\sigma_0 \vee\cdots \vee\sigma_{r-i}\) are sections of
\(\bigwedge\nolimits^{r-i+1} {\mathcal{E}}\to Y\). Then

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  No component of \(Y_{ \sigma}\) has codimension greater than \(i\),
\item
  If the \(\sigma_i\) are general elements of
  \({\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) and
  \(V \subseteq H^0({\mathcal{E}})\) be a subset of global sections
  generating \({\mathcal{E}}\), then \(Y_{ \sigma}\) is generically
  reduced with codimension \(i\) in \(Y\).
\end{enumerate}

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-11-03_14-06.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{lemma}

\begin{remark}

For affine \(Y\), locally thinking of functions, either \(f\) hits or
misses completely any given irreducible component.

\end{remark}

\begin{proof}[of b]

Let \(V\in{\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}\) with
\(\dim_{\mathbb{C}}V = m\) be the module of global sections
\(\sigma_0, \cdots, \sigma_{m-1}\) generating \({\mathcal{E}}\), and let
\(\phi:Y\to {\operatorname{Gr}}_{m-r}(V)\) be given by
\(y \mapsto \ker(V\to {\mathcal{E}}_y)\). If \(W \leq V\) is a submodule
of dimension \(r-i+1\) spanned by \(\sigma_0, \cdots, \sigma_{r-i}\),
then the locus \(Y_{ \sigma} \subseteq Y\) is the preimage
\(\phi^{-1}(X_{\lambda}(W))\).

\end{proof}

\begin{remark}

We can write \(X = {\operatorname{Gr}}_{m-r}(V) = G/P\) for
\(G = \operatorname{GL}(V)\) to realize it as a projective homogeneous
variety. Then
\(X_{\lambda}(W) = \left\{{E \in {\operatorname{Gr}}_{m-r}(V) {~\mathrel{\Big\vert}~}\dim(W \cap E ) \geq 1}\right\}\)
is a Schubert variety for any subspace \(0 \leq W \leq V\). In Young
diagrams for a partition \(\lambda\), this condition corresponds to a
valley:

\includegraphics{figures/2021-11-03_14-22-15.png}

\end{remark}

\begin{theorem}[?]

There is a unique way of assigning to each vector bundle
\({\mathcal{E}}\) on a \(X\) (assumed smooth) a class
\(c({\mathcal{E}}) = 1 + c_1({\mathcal{E}}) + c_2({\mathcal{E}}) + \cdots \in A(X)\),
noting that smooth \(X\) guarantees a ring structure on the Chow group.
These satisfy

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  (Line bundles): If \({\mathcal{L}}\to X\) is a line bundle then
  \(c({\mathcal{L}}) = 1 + c_1({\mathcal{L}})\) where
  \(c_1({\mathcal{L}}) \in A^1(X)\) is the class of the divisor of zeros
  minus the divisor of poles of any rational section of
  \({\mathcal{L}}\), defined up to rational equivalence in \(A(X)\).
\item
  (Degeneracy locus): If \(\sigma_0, \cdots, \sigma_{r-i}\) are global
  sections of \({\mathcal{E}}\) and the degeneracy/dependence locus
  \(Y_{ \sigma} \subseteq X\) has codimension \(i\), then
  \(c_i({\mathcal{E}}) = [X_{ \sigma} ] \in A^i(X)\)
\item
  (Whitney's formula): If
  \(0 \to {\mathcal{E}}\to {\mathcal{F}}\to {\mathcal{G}}\to 0\) is a
  SES in \({ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }_{/ {X}}\), then
  \(c({\mathcal{F}}) = c({\mathcal{E}})c({\mathcal{G}}) \in A(X)\).
\item
  (Functoriality/compatibility with pullback): If \(\phi:X\to Y\) then
  \(\phi^*(c({\mathcal{E}})) = c(\phi^*({\mathcal{E}}))\).
\end{enumerate}

\end{theorem}

\begin{remark}

This induces a map \(c: {\mathsf{K}}(X) \to A(X)\). Note that you can
compose this with the cycle class map
\(A(X)\to H^*_{{\mathrm{sing}}}(X)\).

\end{remark}

\hypertarget{singular-cohomology}{%
\subsection{Singular Cohomology}\label{singular-cohomology}}

\begin{quote}
See Anderson-Fulton.
\end{quote}

\begin{remark}

We can define a total Chern class
\(c({\mathcal{E}}) = \sum_i c_i({\mathcal{E}}) u^i \in R[u]\) for
\(R\coloneqq H^*_{{\mathrm{sing}}}(X)\).

\end{remark}

\begin{proposition}[?]

Setup: take \(X\) paracompact and Hausdorff/T2, which will be necessary
for partitions of unity. For
\({\mathcal{E}}\xrightarrow{\pi} X \in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }({\mathbb{C}})_{/ {X}}\),
there exist \(c_i({\mathcal{E}})\in H^{2i}(X)\) satisfying

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  If \(f:X\to Y \in {\mathsf{Top}}\) then
  \(f^*(c_i({\mathcal{E}})) = c_i(f^* {\mathcal{E}})\).
\item
  \(c_i({\mathcal{E}}) = 0\) unless
  \(o\leq i \leq r\coloneqq\operatorname{rank}({\mathcal{E}})\), and
  \(c_0({\mathcal{E}}) = 1\)
\item
  Exact sequences of vector bundles yield Whitney's formula.
\end{enumerate}

If additionally \(X\) is smooth,

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{3}
\item
  If \({\mathcal{L}}, {\mathcal{M}}\) are line bundles, then
  \(c_1({\mathcal{L}}\otimes{\mathcal{M}}) = c_1({\mathcal{L}}) + c_1({\mathcal{M}})\).
\item
  If \(s:X\to {\mathcal{L}}\) is a nonzero section, writing
  \(Z(s) \subseteq X\) for its zero set,
  \([Z(s)] = c_1({\mathcal{L}}) \in H^2(X)\).
\item
  For the projectivization \(\pi: {\mathbb{P}}({\mathcal{E}})\to X\),
  there is a Poincaré duality: considering
  \({\mathcal{O}}(-1) \subseteq \pi^* {\mathcal{E}}\) and its dual
  \({\mathcal{O}}(1)\),
  \begin{align*}
  H^*({\mathbb{P}}({\mathcal{E}})) = H^*(X)[\zeta] / \left\langle{ \zeta^r - c_1({\mathcal{E}} {}^{ \vee })\zeta^{r-1} + \cdots + (-1)^r c_r({\mathcal{E}} {}^{ \vee }) }\right\rangle
  .\end{align*}
\end{enumerate}

\end{proposition}

\begin{quote}
This is the tautological bundle, to be continued on Friday!
\end{quote}

\hypertarget{friday-november-05}{%
\section{Friday, November 05}\label{friday-november-05}}

\begin{quote}
References: Chriss-Ginzburg (for an introduction), Fulton's
\emph{Intersect Theory} (does a lot).
\end{quote}

\begin{remark}

Today: Borel-Moore homology. For example, characteristic cycles of
\(D{\hbox{-}}\)modules live here. Useful because
e.g.~\(H_*({\mathbb{C}}; {\mathbb{Z}}) = {\mathbb{C}}[0]\), which
doesn't see that \(\dim_{\mathbb{R}}{\mathbb{C}}= 2\). On the other
hand,
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_*({\mathbb{C}}; {\mathbb{Z}}) = {\mathbb{C}}[2]\),
where \(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_*\)
denotes taking Borel-Moore homology. It turns out that if \(X\) is
compact, then
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_* \cong H_*\).

\end{remark}

\begin{definition}[?]

If \(X \hookrightarrow G/P\) be a closed embedding, or more generally
\(X\hookrightarrow M\) for \(M\) any smooth complex manifold (or
quasiprojective variety?) with \(\dim_{\mathbb{C}}M = n\), define
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_k(X) \coloneqq H^{2n-k}(G/P, (G/P)\setminus X)
.\end{align*}

\end{definition}

\begin{remark}

Goal: show this homology contains certain fundamental classes in top
degree
\([X] \in \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n} (X)\).

\end{remark}

\begin{proposition}[?]

There is a group morphism, the \textbf{cycle class map},
\begin{align*}
{ \operatorname{cl}} : A_*(X) \to\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_*(X)
,\end{align*}
such that

\begin{itemize}
\item
  \({ \operatorname{cl}}\) is compatible with proper pushforward,
  i.e.~covariant with respect to proper morphisms. When
  \(X \xrightarrow{f} Y\) is proper, consider the pushforwards
  \(f_*: A_*(X) \to A_*(Y)\) and
  \(f_*' \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_*(X) \to \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_*(Y)\).
  For \(Z \subseteq X\), we can write
  \begin{align*}
  f_*[Z] = 
  \begin{cases}
  d [f(Z)] & { \left.{{f}} \right|_{{Z}} } \text{ degree } d 
  \\
  0 & \text{else}.
  \end{cases}
  .\end{align*}
\item
  \({ \operatorname{cl}}\) is compatible with Chern classes of vector
  bundles.
\end{itemize}

\end{proposition}

\begin{remark}

Fulton sets up \(A_*\) to mimic Borel-Moore homology.

\end{remark}

\begin{lemma}[Existence of fundamental classes]

If \(\dim_{\mathbb{C}}(X) = n\) then
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{>2n}(X) = 0\)
and
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n}(X; {\mathbb{Z}})\)
is a free abelian group with one generator for each irreducible
component of \(X\).

\end{lemma}

\begin{remark}

On restrictions to opens: Let \(U \hookrightarrow X\) be open with
\(X \hookrightarrow G/P\) closed, so that
\(Y\coloneqq X\setminus U \hookrightarrow X\) is closed. Then
\(U \subset (G/P)\setminus Y = (G/P) \setminus(X\setminus U) \subseteq G/P\)
is open. A mnemonic:

\begin{center}
\begin{tikzcd}
    && {G/P} \\
    \\
    X &&&& {(G/P)\setminus Y} \\
    \\
    && U
    \arrow["{\text{closed}}", from=3-1, to=1-3]
    \arrow["{\text{open}}", from=5-3, to=3-1]
    \arrow["{\text{closed}}"', from=5-3, to=3-5]
    \arrow["{\text{open}}"', from=3-5, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwyLCJYIl0sWzIsNCwiVSJdLFs0LDIsIihHL1ApXFxzbSBZIl0sWzIsMCwiRy9QIl0sWzAsMywiXFx0ZXh0e2Nsb3NlZH0iXSxbMSwwLCJcXHRleHR7b3Blbn0iXSxbMSwyLCJcXHRleHR7Y2xvc2VkfSIsMl0sWzIsMywiXFx0ZXh0e29wZW59IiwyXV0=}{Link
to Diagram}
\end{quote}

Then
\begin{align*}
\qty{ (G/P)\setminus Y, (G/P\setminus Y)\setminus U } \subseteq \qty{ G/P, (G/P)\setminus X} 
,\end{align*}
which yields a map
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_k(X) = H^{2n-k}(G/P, (G/P)\setminus X) \to
H^{2n-k}((G/P)\setminus Y, (G/P \setminus Y) \setminus U) = \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_k(U)
.\end{align*}

using that a subvariety of a smooth variety is again smooth of the same
dimension. So we have a map
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_k(X) \to \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_k(U)\),
and this yields a LES:

\end{remark}

\begin{proposition}[LES in Borel-Moore homology]

For \(U \subseteq X\) closed with \(X \subseteq G/P\) and
\(Y\coloneqq X\setminus U\), there is a LES corresponding to
\begin{align*}
(G/P) \setminus X \subseteq (G/P) \setminus Y \subset G/P 
,\end{align*}
given by

\begin{center}
\begin{tikzcd}
    {H^{k+1}(G/P, (G/P)\setminus Y)} && \cdots \\
    \\
    {H^k(G/P, (G/P)\setminus Y)} && {H^k(G/P, (G/P)\setminus X)} && {H^k((G/P)\setminus Y, (G/P)\setminus X)}
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=1-1]
    \arrow[from=1-1, to=1-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwyLCJIXmsoRy9QLCAoRy9QKVxcc20gWSkiXSxbMiwyLCJIXmsoRy9QLCAoRy9QKVxcc20gWCkiXSxbNCwyLCJIXmsoKEcvUClcXHNtIFksIChHL1ApXFxzbSBYKSJdLFswLDAsIkhee2srMX0oRy9QLCAoRy9QKVxcc20gWSkiXSxbMiwwLCJcXGNkb3RzIl0sWzAsMV0sWzEsMl0sWzIsM10sWzMsNF1d}{Link
to Diagram}
\end{quote}

For \(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu\),
this corresponds to

\begin{center}
\begin{tikzcd}
    {\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-k}(Y)} && {\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-k}(X)} && {\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-k}(U)} \\
    \\
    {\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-k-1}(Y)} && \cdots
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-5, to=3-1]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-1, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXGJhcntIfV97Mm4ta30oWSkiXSxbMiwwLCJcXGJhcntIfV97Mm4ta30oWCkiXSxbNCwwLCJcXGJhcntIfV97Mm4ta30oVSkiXSxbMCwyLCJcXGJhcntIfV97Mm4tay0xfShZKSJdLFsyLDIsIlxcY2RvdHMiXSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw0XSxbMyw0XV0=}{Link
to Diagram}
\end{quote}

\end{proposition}

\begin{remark}

Recall that for \(B \subseteq A \subseteq X\), we got an inclusion of
pairs \((A, B) \subseteq (X, B) \subseteq (X, A)\). Also note that we
used
\begin{align*}
(G/P) \setminus X = ( (G/P) \setminus Y)\setminus U
\end{align*}
where \(Y \coloneqq X\setminus U\).

\end{remark}

\begin{proof}[of lemma, there exist fundamental classes]

Let \(Y\) be the singular locus of \(X\) with
\(n\coloneqq\dim_{\mathbb{C}}X\), then

\begin{itemize}
\tightlist
\item
  \(Y \subseteq X\) is closed, and
\item
  \(\dim_{\mathbb{C}}Y < \dim_{\mathbb{C}}X\) is strictly smaller.
\end{itemize}

Strategy: induct on \(\dim X\) and use the LES applied to
\(U \coloneqq X\setminus Y\) and \(Y\). Note that \(U\) is smooth. We
have
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n}(Y) \to \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n}(X) \to \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n}(U) \xrightarrow{\delta} \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-1}(Y)
,\end{align*}
and
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n}(Y) = 0\)
since \(\dim Y < 2n\) and
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-1}(Y) = 0\)
for the same reason, making the middle map an isomorphism. Write
\(U = \displaystyle\coprod_{0\leq i\leq \ell} U_i\) as a union of
irreducible (so connected) components. Then
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n}(U) = H^{2n-2n}(U, U \setminus U) = H^0(U) = {\mathbb{Z}}{ {}^{ \scriptscriptstyle\oplus^{\ell} }  }
\end{align*}
where we can choose to embed \(U \hookrightarrow U\) into itself since
\(U\) is smooth. Any Zariski open has to intersect every irreducible
component, so each such component yields a fundamental class.

\end{proof}

\hypertarget{monday-november-08}{%
\section{Monday, November 08}\label{monday-november-08}}

\begin{remark}

Today: Poincaré duality, relates to \emph{smooth} loci, and rational
(i.e.~\({\mathbb{Q}}\)) smoothness. Take all varieties to be
quasiprojective subvarieties of a flag variety \(G/B\).

From algebraic topology, there is a relative cup product in singular
cohomology:
\begin{align*}
\smile: H^i(X, U; {\mathbb{Z}}) \otimes_{\mathbb{Z}}H^j(X, U; {\mathbb{Z}}) \to H^{i+j}(X, U_1\cup U_2)
.\end{align*}
Even better, we have a pairing with Borel-Moore homology for
\(Y \leq X\) a closed subvariety:
\begin{align*}
\frown: H^j(X, X\setminus Y) \otimes_{\mathbb{Z}}\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_j(X) &\to \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{j-i}(Y)
.\end{align*}
This yields
\begin{align*}
\frown: H^j(X, X\setminus Y) \otimes H^{2n-j}(G/P, (G/P)\setminus X) \to H^{2n-j+i}(G/P, (G/P) \setminus Y)
.\end{align*}
Think of \(H^j(A, B)\) as chains in \(A\) vanishing along \(B\):

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-11-08_14-01.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\end{remark}

\begin{proposition}[Poincaré duality]

For \(X\) smooth and irreducible, capping against the fundamental class
induces an isomorphism
\begin{align*}
H^i(X) &{ { \, \xrightarrow{\sim}\, }}H_{2n-i}(X) \\
\alpha &\mapsto \alpha \frown[X]
,\end{align*}
which is induced by
\begin{align*}
H^i(X) \times \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n}(X) &\to \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-i}(X) \\
(\alpha, [X]) &\mapsto \alpha \frown[X]
.\end{align*}

\end{proposition}

\begin{remark}

Recall that there is an affine stratification
\(G/P_Y = \displaystyle\coprod_{w\in W^Y} BwP_Y/P_Y\), and
\begin{align*}
\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2k} (G/P) = \bigoplus_{\substack{ w\in W^Y \\ \ell(w) = k }} {\mathbb{Z}}[X_w^Y]
.\end{align*}
Pulling back along the isomorphism there is some element such that
\(d_{X_w^Y} \frown[G/P] = [X_w^Y]\), so we often identify
\(d_{X_w^Y} = [X_w^Y]\).

\end{remark}

\begin{remark}

An alternative perspective on Chern classes: compose the maps

\begin{center}
\begin{tikzcd}
    {{\mathsf{K}}(X)} && {A(X)} && {\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_*(X)} && {H^*(X)} \\
    \\
    {\varepsilon}&& {c_1({\varepsilon})} && {c_1({\varepsilon})} && {c_1({\varepsilon})} \\
    && {[Z(s)]} && {[Z(s)]} && {[Z(s)]}
    \arrow[from=1-1, to=1-3]
    \arrow[from=1-3, to=1-5]
    \arrow[from=1-5, to=1-7]
    \arrow[from=3-1, to=3-3]
    \arrow[from=3-3, to=3-5]
    \arrow[from=3-5, to=3-7]
    \arrow[from=4-3, to=4-5]
    \arrow[from=4-5, to=4-7]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTEsWzAsMCwiXFxLKFgpIl0sWzIsMCwiQShYKSJdLFs0LDAsIlxcYmFye0h9XyooWCkiXSxbNiwwLCJIXiooWCkiXSxbMCwyLCJcXGVwcyJdLFsyLDIsImNfMShcXGVwcykiXSxbNCwyLCJjXzEoXFxlcHMpIl0sWzYsMiwiY18xKFxcZXBzKSJdLFsyLDMsIltaKHMpXSJdLFs0LDMsIltaKHMpXSJdLFs2LDMsIltaKHMpXSJdLFswLDFdLFsxLDJdLFsyLDNdLFs0LDVdLFs1LDZdLFs2LDddLFs4LDldLFs5LDEwXV0=}{Link
to Diagram}
\end{quote}

Here \(Z(s)\) is the zero divisor of a section \(s\) coming from the
class of a bundle in \({\mathsf{K}}(X)\). For a line bundle
\({\mathcal{L}}\), we have
\(c_1({\mathcal{L}}) \in A^1(X) { { \, \xrightarrow{\sim}\, }}\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_{2n-2} { { \, \xrightarrow{\sim}\, }}H^2(X)\).

\end{remark}

\begin{theorem}[On nilpotent orbits, Borho-MacPherson]

TFAE:

\begin{itemize}
\tightlist
\item
  \(H^i(X, X\setminus\left\{{x}\right\} ) = {\mathbb{Q}}[2n]\)
\item
  \({\mathbb{R}}{{\Gamma}\qty{X, \operatorname{\mathcal{IC}}_X} } = {\mathbb{Q}}[0]\)
\end{itemize}

\end{theorem}

\begin{remark}

Mentioned by Geordie:
\(\operatorname{\mathcal{IC}}_X \cong {\mathbb{Q}}_X\), the constant
sheaf.

\end{remark}

\begin{example}[?]

Let
\begin{align*}
f(x,y,z) = x^3 + y^3 - xyz
.\end{align*}
Let \(X\coloneqq V(f) \subseteq {\mathbb{P}}^2_{/ {{\mathbb{C}}}}\), and
define
\begin{align*}
\xi: {\mathbb{P}}^1_{/ {{\mathbb{C}}}} &\to X \\
[a: b] &\mapsto [ab^2: a^2b : a^3 + b^3]
.\end{align*}

Check that this is well-defined:
\begin{align*}
(ab^2)^3 + (a^2b)^3 - a^3b^3 (a^3 + b^3)
.\end{align*}
Note \(\xi\) is projective and thus proper, and finite since it is
quasifinite (finite fibers). One can check
\begin{align*}
\xi^{-1}[0:0:1] &= \left\{{ [0:1], [1:0] }\right\} \\
\xi^{-1}[x:y:z] &= \left\{{{\operatorname{pt}}}\right\}
.\end{align*}

\begin{exercise}[?]

Check that \(\xi\) is birational.

\end{exercise}

Thus \(\xi\) is the normalization of \(X\), but isn't an isomorphism, so
smoothness must fail.

\begin{question}

Is \(X\) \emph{rationally} smooth?

\end{question}

Since \(X\) is compact,
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_k(X) { { \, \xrightarrow{\sim}\, }}H_k(X)\).
Since \(X\) is connected we get \(H^0 = {\mathbb{Q}}\), and by duality
\(H^2(X) \cong H_2(X) \cong \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu_2(X) \cong {\mathbb{Q}}\),
we have
\(H^k(X) = {\mathbb{Q}}[0] \oplus {\mathbb{Q}}[1] \oplus {\mathbb{Q}}[2]\).
Note that the Poincaré polynomial \(p(q) = 1 + q + q^2\) has symmetric
coefficients. What this morphism looks like:

\begin{figure}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/FlagVarieties/sections/figures}{2021-11-08_14-25.pdf_tex} };
\end{tikzpicture}
}
\end{figure}

\begin{claim}

\(X\) is not rationally smooth.

\end{claim}

\begin{proof}[?]

By the projection formula,
\begin{align*}
\xi_*(\xi^* \alpha\frown\beta) = \alpha\frown\xi_* \beta
.\end{align*}
Let \(\alpha \in H^1(X)\) be nonzero, then
\begin{align*}
\alpha\frown[X] = a \frown\xi_* [ {\mathbb{P}}^1] = \xi_*(\xi^* \alpha \frown[{\mathbb{P}}^1] )
.\end{align*}
Since \(\xi\) is birational, \(\xi_* [{\mathbb{P}}^1] = [X]\) and
\(H^k({\mathbb{P}}^1) = {\mathbb{Q}}[0] \oplus {\mathbb{Q}}[2]\).
Rationally smooth implies PD, and since PD doesn't hold here we can't
have rational smoothness.

\end{proof}

\end{example}

\hypertarget{friday-november-12}{%
\section{Friday, November 12}\label{friday-november-12}}

\begin{example}[Projective space]

Let
\(G\curvearrowright X \in {\mathsf{{\mathbb{C}}}{\hbox{-}}\mathsf{Mod}}^{\dim = n}\)
be a linear algebraic group acting on a \({\mathbb{C}}{\hbox{-}}\)module
of dimension \(n\), then there is a morphism
\(G\to \operatorname{GL}_n\) and we'll regard
\(G \subseteq \operatorname{GL}_n\). Then
\(G\curvearrowright{\mathbb{P}}^n\):

\begin{itemize}
\tightlist
\item
  \({\mathbb{P}}(V) = \leftquotient{{\mathbb{C}}^{\times}}{V\setminus\left\{{0}\right\}}\),
  and \(G\) acts linearly and commutes with scalar multiplication.
\item
  \({\mathbb{P}}(V) = \operatorname{GL}_n / P\) and the
  \(G{\hbox{-}}\)action descends since the projection
  \(\operatorname{GL}_n\to \operatorname{GL}_n/P\) is
  \(\operatorname{GL}_n{\hbox{-}}\)equivariant.
\end{itemize}

Note that \(G\) also acts on the tautological bundle
\({\mathcal{O}}(-1)\), since these are lines. We can write
\({\mathcal{O}}(-1) = \operatorname{GL}_n \overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_{{\left[ {1,0,\cdots, 0} \right]}}\),
using the identification
\(X^*(T) = {\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{n} } }\) and
taking the character associated to
\({\left[ {t_1,\cdots, t_n} \right]}\mapsto t_1\). Note that
\({\mathcal{O}}(-1) \to\operatorname{GL}_n/P\) is
\(\operatorname{GL}_n\) equivariant. Write
\(\zeta \coloneqq c_1^G({\mathcal{O}}(1)) \in H_G^2({\mathbb{P}}(V))\)
for the equivariant Chern classes.

Recall that if
\(\operatorname{GL}_n \overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_{\lambda} \to \operatorname{GL}_n/P\)
for \(\lambda \in X^*(T)\) is a \(G{\hbox{-}}\)equivariant bundle, we
can construct
\(E\overset{\scriptscriptstyle {G} }{\times} \operatorname{GL}_n\overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_{\lambda} \to E\overset{\scriptscriptstyle {G} }{\times} \operatorname{GL}_n/P \cong E\overset{\scriptscriptstyle {G} }{\times} \times {\mathbb{P}}^{n-1}\),
and the base here corresponds to \(H^*_G({\mathbb{P}}^{n-1})\). This
induces
\(E\overset{\scriptscriptstyle {G} }{\times} {\mathbb{P}}^{n-1} \to E/G = {{\mathbf{B}}G}\),
where now the base corresponds to \(H^*_G({\operatorname{pt}})\).

\end{example}

\begin{proposition}[?]

\begin{align*}
H^*_G({\mathbb{P}}(V)) { { \, \xrightarrow{\sim}\, }}H^*_G({\operatorname{pt}})[\zeta] / \left\langle{\sum_{k=0}^n c_k \zeta^{n-k}}\right\rangle
.\end{align*}

\end{proposition}

\begin{proof}[?]

Given \({\mathcal{E}}\to X\), we know
\(H^*({\mathbb{P}}({\mathcal{E}}))\) in terms of
\(H^*({\operatorname{pt}})\). We have

\begin{center}
\begin{tikzcd}
    {E\overset{\scriptscriptstyle {G} }{\times} \operatorname{GL}_n/P} && {{\mathbb{P}}(E\overset{\scriptscriptstyle {G} }{\times} \operatorname{GL}_n \overset{\scriptscriptstyle {P} }{\times} {\mathbb{C}}_\lambda)} \\
    \\
    {E/G = {{\mathbf{B}}G}} && {{\mathbf{B}}G}
    \arrow["{=}", from=1-1, to=1-3]
    \arrow[from=1-1, to=3-1]
    \arrow["{=}"', from=3-1, to=3-3]
    \arrow[from=1-3, to=3-3]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJFXFxtaXh7R30gXFxHTF9uL1AiXSxbMCwyLCJFL0cgPSBcXEJHIl0sWzIsMCwiXFxQUChFXFxtaXh7R30gXFxHTF9uIFxcbWl4e1B9IFxcQ0NfXFxsYW1iZGEpIl0sWzIsMiwiXFxCRyJdLFswLDIsIj0iXSxbMCwxXSxbMSwzLCI9IiwyXSxbMiwzXV0=}{Link
to Diagram}
\end{quote}

So \(\xi\) is a hyperplane class for a projective bundle, and thus
\(c_i^G(V) = c_i(E\overset{\scriptscriptstyle {G} }{\times} V)\).

\end{proof}

\begin{example}

For \(G = \operatorname{GL}_n\), we have
\(H^*_G({\operatorname{pt}}) = {\mathbb{Z}}[c_1,\cdots, c_n] \subseteq {\mathbb{Z}}[t_1,\cdots, t_n] = H^*_T({\operatorname{pt}})\).
So
\(H^*_G {\mathbb{P}}(V) = {\mathbb{Z}}[c_1,\cdots, c_n][\zeta]/\left\langle{\zeta^n + c_1\zeta^{n-1} + \cdots + c_n}\right\rangle\).

\end{example}

\begin{example}[?]

For \(G=T\),
\(H^*_T({\mathbb{P}}(V)) = {\mathbb{Z}}[t_1,\cdots, t_n][\zeta] / \left\langle{\prod_{1\leq i \leq n} \zeta + t_i}\right\rangle\)
where \(c_i^T(V) = e_2(t_1,\cdots, t_n)\).

\end{example}

\begin{theorem}[Localization in equivariant cohomology]

Let \(X\) be an \(n{\hbox{-}}\)dimensional smooth algebraic variety with
finitely many \(T{\hbox{-}}\)fixed points. Write \(X^T\) for the fixed
point locus, write
\(c \coloneqq\prod_{p\in X^T} c_n^T({\mathbf{T}}_p X) \in H^*_T({\operatorname{pt}})\),
noting that since \(X\) is smooth these are all the same dimension. Let
\(S \subseteq H^*_T({\operatorname{pt}})\) be a multiplicative set
containing \(c\), which is nonzero since the fixed points are isolated.
Assume there are \(m\leq {\sharp}X^T\) classes in \(H_T^*(X)\)
restricting to a basis of \(H^*(X)\). Then there are isomorphisms
induced by
\begin{align*}
H^*_T(X) \left[ { \scriptstyle { {S}^{-1}}  } \right] &\xrightarrow{S^{-1}i^*} H^*_T(X^T) \left[ { \scriptstyle { {S}^{-1}}  } \right] \\
H^*_T(X^T) \left[ { \scriptstyle { {S}^{-1}}  } \right] &\xrightarrow{S^{-1}i_*} H^*_T(X) \left[ { \scriptstyle { {S}^{-1}}  } \right]
.\end{align*}
Note that \(X^T \xrightarrow{i} X\) is \(T{\hbox{-}}\)equivariant, so
\(i^*\) on \(H^*\) descends to \(H_T^*\). By Poincaré duality, we get
\(\mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu(X^T) \to \mkern 1.5mu\overline{\mkern-1.5muH\mkern-1.5mu}\mkern 1.5mu(X)\).
Without the localization, there is still an injection:
\begin{align*}
H_T^*(X) \xrightarrow{\iota^*} H^*_T(X^T) = \bigoplus H^*_T({\operatorname{pt}})
.\end{align*}

\end{theorem}

\begin{remark}

Note that
\(H^*_T({\operatorname{pt}}; {\mathbb{Z}}) = {\mathbb{Z}}[t_1,\cdots, t_n]\)
and
\(H^*({\operatorname{pt}}; {\mathbb{C}}) = {\mathbb{C}}[t_1,\cdots, t_n] = S({\mathfrak{h}} {}^{ \vee })\),
the symmetric algebra on the Cartan. This comes up when looking at
Soergel bimodules. Compare to
\({\mathsf{K}}^T({\operatorname{pt}}) = R(T)\), the representation ring.

\end{remark}

\begin{example}[?]

For projective space, let \(T\) be any torus that acts linearly on a
\(n{\hbox{-}}\)dimensional \({\mathbb{C}}{\hbox{-}}\)module. Then
\(V = \bigoplus_i C_{\lambda_i}\) for some characters \(\lambda_i\).
Assume the \(\lambda_i\) are distinct, then
\begin{align*}
H^*_T({\mathbb{P}}^{n-1}) = H^*_T({\operatorname{pt}})[\zeta]/\left\langle{\prod \zeta + \lambda_i}\right\rangle
,\end{align*}
where \(\zeta = c_1^T({\mathcal{O}}(1))\). So write \(X^T\) as the set
of coordinate lines for \(X = {\mathbb{P}}^{n-1} = {\mathbb{P}}(V)\),
i.e.~for \(p_i\coloneqq{\left[ {0,0,\cdots, 0,1,0,\cdots,0} \right]}\),
\(X^T = \left\{{p_1,\cdots, p_n}\right\}\). The tangent spaces are given
by
\({\mathbf{T}}_{p_i} {\mathbb{P}}^{n-1} = \bigoplus_{j\neq i} {\mathbb{C}}_{\lambda_j - \lambda_i} = {\mathbf{T}}_{p_i} U_i\)
where \(U_i \cong {\mathbb{C}}^{n-1}\) by dividing out by the \(i\)th
coordinate, so
\begin{align*}
t{\left[ {x_1: \cdots : 1:\cdots x_n} \right]} 
&= {\left[ {t_1x_1: \cdots : t_i \cdot 1: \cdots t_n x_n } \right]} \\
&= {\left[ {{t_1\over t_i} x_1 : \cdots: 1 : \cdots {t_n \over t_i} x_n} \right]}
.\end{align*}
Thus \((\lambda_j - \lambda_i)(t) = t_1/t_i\). Thus
\(c_{n-1}^T({\mathbf{T}}_{p_i} {\mathbb{P}}^{n-1}) = \prod_{j\neq i} (\lambda_j - \lambda_i)\).

\end{example}

\begin{proposition}

A self-intersection formula: if \(i:Y\hookrightarrow X\) is a closed
embedding of codimension \(d\) with normal bundle \(N\) of rank \(d\),
then
\begin{align*}
i^* i_* ( \alpha) = c_d(N) \alpha
.\end{align*}

\end{proposition}

\begin{exercise}[?]

Show that the following composite is diagonal:
\begin{align*}
H_T{ {}^{ \scriptscriptstyle\oplus^{n} }  } \to H_T({\mathbb{P}}^{n-1}) \to H_T{ {}^{ \scriptscriptstyle\oplus^{n} }  }
.\end{align*}
What is the determinant?

\end{exercise}

\hypertarget{monday-november-22}{%
\section{Monday, November 22}\label{monday-november-22}}

\begin{remark}

Considering the infinite dimensional case, \(\tilde A_2\). Here
\(W = W(\tilde A_2) = \left\langle{s_1,s-2,s_3 {~\mathrel{\Big\vert}~}s_i^2=1, (s_i s_j)^3=1}\right\rangle\),
and we can form \(X_w \subseteq G/B\) for any \(w\in W\). This will be a
finite dimensional projective variety with a Torus action, and there are
BSDH resolutions for reduced words given by \(T{\hbox{-}}\)equivariant
maps
\begin{align*}
P_{i_1}\overset{\scriptscriptstyle {B} }{\times} \cdots P_{i_n}/B \xrightarrow{\mu} X_w
.\end{align*}
These are resolutions of singularities, and in particular birational.
Note that \(W\) is infinite here.

\end{remark}

\begin{remark}

Article by Graham-Li: say \(w\in W\) is \textbf{spiral} iff
\(w = (s_j s_j s_k)^\ell\) for \(i,j,k \in \left\{{1,2,3}\right\}\).
This produces a nice family of Schubert varieties. For
\(\operatorname{rank}A = 2\), we have \(\dim {\mathfrak{h}}= 3+1=4\). Up
to a change of coordinates, we can use
\(\alpha_1 {}^{ \vee }= {\left[ {1,0} \right]}^t\) and
\(\alpha_2 {}^{ \vee }= {\left[ {1,0} \right]}^t\) and let
\(V \coloneqq R \otimes_{\mathbb{Z}}\left\{{\alpha_1 {}^{ \vee }, \alpha_2 {}^{ \vee }}\right\}\)
be the ambient Euclidean space and set
\(L\coloneqq{\mathbb{Z}}\left\langle{ \alpha_1, \alpha_2}\right\rangle\).
Then use the action of \(W\) to define
\(W_{\text{aff}} \coloneqq L \rtimes W_f\) where
\(W_f = W(A {}^{ \vee })\), and
\(s_i(\chi) = \chi - {\left\langle { \alpha, \lambda},~{\alpha} \right\rangle} {}^{ \vee }\).
Here we think of \(L\) as translations. The dual roots are
\(\alpha_1 = {\left[ {2, -1} \right]}^t. \alpha_2 = {\left[ {-1, 2} \right]}\)
and so
\(\tilde \alpha = \alpha_1 + \alpha_2 = {\left[ {1,1} \right]}^t\)
Define hyperplanes
\(H_{\alpha, n} \coloneqq\left\{{v\in V {~\mathrel{\Big\vert}~}{\left\langle {\alpha},~{v} \right\rangle} = n \in {\mathbb{Z}}}\right\}\).
There is a \emph{fundamental alcove} enclosed by the positive sides of
the various hyperplanes and within distance 1 of
\(H_{\tilde \alpha, 0}\). If you draw the picture and now act on the
fundamental alcove by simple reflections, the image ``spirals'' out away
from the origin.

\end{remark}

\begin{remark}

The article doesn't use BSDH resolutions, maybe compare and contrast
with what we've done.

\end{remark}

\begin{remark}

Back to \(\mu\). For \(\ell = 1\), we have
\begin{align*}
P_1 \overset{\scriptscriptstyle {B} }{\times} P_2 \overset{\scriptscriptstyle {B} }{\times} P_3/B \xrightarrow{\mu} X_w
.\end{align*}

The Bruhat order yields

\begin{center}
\begin{tikzcd}
    && {s_1s_2s_3} \\
    {s_1s_2} && {s_1s_3} && {s_2s_3} \\
    \\
    {s_1} && {s_2} && {s_3} \\
    \\
    && 1
    \arrow[from=4-1, to=6-3]
    \arrow[from=4-3, to=6-3]
    \arrow[from=4-5, to=6-3]
    \arrow[from=2-1, to=4-1]
    \arrow[from=2-3, to=4-3]
    \arrow[from=2-5, to=4-5]
    \arrow[from=2-1, to=4-3]
    \arrow[from=2-3, to=4-1]
    \arrow[from=2-3, to=4-5]
    \arrow[from=2-5, to=4-3]
    \arrow[from=1-3, to=2-1]
    \arrow[from=1-3, to=2-3]
    \arrow[from=1-3, to=2-5]
\end{tikzcd}
\end{center}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMiwwLCJzXzFzXzJzXzMiXSxbMCwxLCJzXzFzXzIiXSxbMiwxLCJzXzFzXzMiXSxbNCwxLCJzXzJzXzMiXSxbMCwzLCJzXzEiXSxbMiwzLCJzXzIiXSxbNCwzLCJzXzMiXSxbMiw1LCIxIl0sWzQsN10sWzUsN10sWzYsN10sWzEsNF0sWzIsNV0sWzMsNl0sWzEsNV0sWzIsNF0sWzIsNl0sWzMsNV0sWzAsMV0sWzAsMl0sWzAsM11d}{Link
to Diagram}
\end{quote}

Note that there are no braid relations. We can consider the
\(T{\hbox{-}}\)equivariant multiplicity
\({\mathcal{E}}_x^T X_w = \sum_{z\in \mu^{-1}(x)^T} {\mathcal{E}}_z^T(z)\)
given by summing over the \(T{\hbox{-}}\)equivariant fixed points in the
fiber. Here this just equals \({\mathcal{E}}_z^T(z)\) where
\(\mu(z) = x\), since there is a unique \(T{\hbox{-}}\)fixed point in
the fiber. A basic AG argument shows that the resolution is an
isomorphism and thus \(X_w\) is smooth, so there is no singular locus.
The paper gives a nice formula for \(\ell \geq 6\).

\end{remark}

\begin{remark}

Starting the calculation:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Consider
  \(e_x X_w\in \operatorname{ff}(S({\mathfrak{h}} {}^{ \vee }))\) the
  equivariant multiplicity, then \(x\in X_w\) is smooth iff a certain
  change of basis \(c_{w, x}\) corresponds to the equivariant
  multiplicity.
\item
  In the rationally smooth locus, they show smoothness iff there is a
  single \(T{\hbox{-}}\)fixed point in the fiber.
\end{enumerate}

\end{remark}

\hypertarget{sashas-talk-monday-november-29}{%
\section{Sasha's Talk (Monday, November
29)}\label{sashas-talk-monday-november-29}}

\begin{remark}

Topic: Segal-Sagawara construction. Define
\(\mathrm{Witt} = \operatorname{Lie}(\operatorname{Diff}^+ S^1)\),
regarded as polynomial vector fields on \(S^1\).
\(H^2(\mathrm{Witt}; {\mathbb{C}}) = {\mathbb{C}}\), so there is a
1-dimensional space of central extensions, with a distinguished one: the
Virasoro algebra. There is a SES
\(0 \to{\mathbb{C}}\mathrm{charge} \to \mathrm{Vir} \to \mathrm{Witt}\to 0\),
and for \(LG\coloneqq C^\infty(S^1, G)\), a SES
\(0\to S^1 \to \tilde{LG} \to LG\to 0\). Here \(\mathrm{charge}\) is
some distinguished central element. Does the Virasoro group act on this
extension? Not quite, but almost -- pass to Lie algebras to get
\(0\to {\mathbb{C}}\to \tilde{L{\mathfrak{g}}}\to {\mathfrak{g}}\to 0\).
Theorem: for
\(\rho: \tilde{L{\mathfrak{g}}} \to \mathop{\mathrm{End}}_{\mathbb{C}}(V)\)
an admissible representation, there is a representation
\(\rho': \mathrm{Vir}\to \mathop{\mathrm{End}}_{\mathbb{C}}(V)\). Note
\(L{\mathfrak{g}}\coloneqq{\mathfrak{g}}\otimes_{\mathbb{C}}{\mathbb{C}}[t, t^{-1}]\).
Write \(X_i \left\langle{m}\right\rangle \coloneqq X_i\otimes m\).

\end{remark}

\begin{remark}

Admissible representations: for all \(v\in V\) and
\(X\in{\mathfrak{g}}\), there exists an \(m\) such that
\(\rho(X\left\langle{m}\right\rangle)(v) = 0\). Define the Casimir
element \(\sum_i X_i X^i \in Z({\mathcal{U}}({\mathfrak{g}}))\). Levels:
level \(\ell\) if \(\mathrm{charge}\) acts by
\(\ell \cdot \operatorname{id}\). Critical level: \(\ell \neq \cdots\)
some constant (roughly the dual Coxeter number), avoid this \(\ell\) for
the reps in the theorem statement.

\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
  \({\mathbf{B}}G\)

  \begin{itemize}
  \tightlist
  \item
    Some examples?
    \({\mathbb{CP}}^\infty, {\mathbf{B}}\operatorname{GL}_n({\mathbb{R}})\),
    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}