lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 1dbcc654c5fc0b8868792f0f00e56f5db5ec8ec4
Parent: 7179f4432b3238fb409deb2a13ecfc60638d713a
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 18 Sep 2022 20:46:07 +0000

Started to work on the chapter about Mathieu's paper

Diffstat

3 files changed, 168 insertions, 2 deletions

Status	File Name	N° Changes	Insertions	Deletions
Added	sections/mathieu.tex	164	164	0
Modified	sections/semisimple-algebras.tex	4	2	2
Modified	tcc.tex	2	2	0

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -0,0 +1,164 @@
+\chapter{Irreducible Weight Modules}
+
+\begin{definition}
+  A representation \(V\) of \(\mathfrak{g}\) is called a \emph{weight
+  \(\mathfrak{g}\)-module} if \(V = \bigoplus_{\lambda \in \mathfrak{h}^*}
+  V_\lambda\) and \(\dim V_\lambda < \infty\) for all \(\lambda \in
+  \mathfrak{h}^*\). The \emph{support of \(V\)} is the set
+  \(\operatorname{supp} V = \{\lambda \in \mathfrak{h}^* : V_\lambda \ne 0\}\).
+\end{definition}
+
+\begin{definition}
+  A weight \(\mathfrak{g}\)-module is called
+  \emph{an admissible} if \(\dim V_\lambda\) is bounded. The lowest upper bound
+  for \(\dim V_\lambda\) is called \emph{the degree of \(V\)}.
+\end{definition}
+
+\begin{example}
+  Corollary~\ref{thm:finite-dim-is-weight-mod} is equivalent to the fact that
+  every finite-dimensional \(\mathfrak{g}\)-module is a weight module.
+\end{example}
+
+% TODO: Is every quotient of a weight module a weight module too?
+\begin{example}
+  Proposition~\ref{thm:verma-is-weight-mod} and
+  proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
+  \(M(\lambda)\) and its maximal subrepresentation are both weight modules. In
+  fact, the proof of proposition~\ref{thm:max-verma-submod-is-weight} is
+  actually a proof of the fact that every subrepresentation of a weight module
+  is a weight module.
+\end{example}
+
+\begin{definition}
+  A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
+  if \(\mathfrak{b} \subset \mathfrak{p}\).
+\end{definition}
+
+% TODO: Comment afterwords that the Verma modules are indeed generalized Verma
+% modules
+\begin{definition}
+  Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) and a
+  \(\mathfrak{p}\)-module \(V\) the module \(M_{\mathfrak{p}}(V) =
+  \operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is called \emph{a
+  generalized Verma module}.
+\end{definition}
+
+\begin{proposition}
+  Given an irreducible \(\mathfrak{p}\)-module \(V\), the generalized Verma
+  module \(M_{\mathfrak{p}}(V)\) has a unique maximal subrepresentation
+  \(N_{\mathfrak{p}}(V)\) and a unique irreducible quotient
+  \(L_{\mathfrak{p}}(V) = \mfrac{M_{\mathfrak{p}}(V)}{N_{\mathfrak{p}}(V)}\).
+  The irreducible quotient \(L_{\mathfrak{p}}(V)\) is a weight module.
+\end{proposition}
+
+\begin{definition}
+  An irreducible \(\mathfrak{g}\)-module is called \emph{parabolic induced} if
+  it is isomorphic to \(L_{\mathfrak{p}}(V)\) for some proper parabolic
+  subalgebra \(\mathfrak{p} \subsetneq \mathfrak{g}\) and some
+  \(\mathfrak{p}\)-module \(V\). An \emph{irreducible cuspidal
+  \(\mathfrak{g}\)-module} is an irreducible representation of \(\mathfrak{g}\)
+  which is \emph{not} parabolic induced.
+\end{definition}
+
+% TODO: Remark on the fact that any simple weight p-mod is a (p/u)-mod, so that
+% the notation of a cuspidal p-mod is well definited
+% TODO: Define the conjugation of a p-mod by an element of the Weil group
+\begin{theorem}[Fernando]
+  Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to
+  \(L_{\mathfrak{p}}(V)\) for some parabolic subalgebra \(\mathfrak{p} \subset
+  \mathfrak{g}\) and some irreducible cuspital \(\mathfrak{p}\)-module \(V\).
+  Furthermore, if \(\mathfrak{p}_1, \mathfrak{p}_2 \subset \mathfrak{g}\) are
+  parabolic and \(V_i\) is an irreducible cuspidal \(\mathfrak{p}_i\)-module
+  then \(L_{\mathfrak{p}_1}(V_1) \cong L_{\mathfrak{p}_2}(V_2)\) if, and only
+  if \(\mathfrak{p}_1 = \mathfrak{p}_2^w\) and \(V_1 \cong V_2^w\) for some \(w
+  \in W\).
+\end{theorem}
+
+% TODO: Remark that the support of a simple weight module is always contained
+% in a coset
+% TODO: Note that conditions (ii) and (iii) have special names
+\begin{corollary}[Fernando]
+  Let \(V\) be an irreducible weight \(\mathfrak{g}\)-module. The following
+  conditions are equivalent.
+  \begin{enumerate}
+    \item \(V\) is cuspidal.
+    \item \(E_\alpha\) acts injectively in \(V\) for all \(\alpha \in \Delta\).
+    \item The support of \(V\) is precisely one \(Q\)-coset.
+  \end{enumerate}
+\end{corollary}
+
+\begin{proposition}
+  If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
+  \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
+  and \(\mathfrak{s}_i\) is a simple component of \(\mathfrak{g}\), then any
+  irreducible weight \(\mathfrak{g}\)-module \(V\) decomposes as 
+  \[
+    V = Z \otimes V_1 \otimes \cdots \otimes V_n
+  \]
+  where \(Z\) is a 1-dimensional representation of \(\mathfrak{z}\) and \(V_i\)
+  is an irreducible weight \(\mathfrak{s}_i\)-module.
+\end{proposition}
+
+\begin{definition}
+  A \emph{coherent family \(\mathcal{M}\) of degree \(d\)} is a weight
+  \(\mathfrak{g}\)-module \(\mathcal{M}\) such that
+  \begin{enumerate}
+    \item \(\dim \mathcal{M}_\lambda = d\) for \emph{all} \(\lambda \in
+      \mathfrak{h}^*\)
+    \item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the centralizer
+      \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) of \(\mathfrak{h}\) in
+      \(\mathcal{U}(\mathfrak{g})\), the map
+      \begin{align*}
+        \mathfrak{h}^* & \to K \\
+               \lambda & \mapsto
+               \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\lambda})
+      \end{align*}
+      is polynomial in \(\lambda\).
+  \end{enumerate}
+\end{definition}
+
+\begin{definition}
+  A coherent family \(\mathcal{M}\) called \emph{irreducible} if
+  \(\mathcal{M}_\lambda\) is a simple
+  \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module for each \(\lambda \in
+  \mathfrak{h}^*\).
+\end{definition}
+
+\begin{definition}
+  Given a representation \(V\) of \(\mathfrak{g}\), a coherent extension
+  \(\mathcal{M}\) of \(V\) is a coherent family \(\mathcal{M}\) that contains
+  \(V\) as a subquotient.
+\end{definition}
+
+% TODO: Define the semisimplification of a coherent family
+\begin{theorem}[Mathieu]
+  Let \(V\) be an infinite-dimensional admissible irreducible
+  \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
+  cohorent extension \(\operatorname{Ext}(V)\) of \(V\). The central characters
+  of the irreducible submodules of \(\operatorname{Ext}(V)\) are all the same.
+  Furthermore, if \(\mathcal{M}\) is any coherent extension of \(V\), then
+  \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
+\end{theorem}
+
+% TODO: Add a proof!
+% TODO: V is a submodule because of the definition of the semisimplification
+% TODO: All weights in the coset occur because V is cuspidal
+% TODO: The dimension of the weight spaces of V is maximal because Ext is
+% an irreducible coherent family
+% TODO: Define the notation for M[mu] somewhere else
+\begin{proposition}
+  Let \(V\) be a cuspidal representation of \(\mathfrak{g}\) and take any
+  weight \(\lambda\) of \(V\). Then \(V \cong
+  (\operatorname{Ext}(V))[\lambda]\).
+\end{proposition}
+
+\begin{theorem}[Mathieu]
+  Let \(\mathcal{M}\) be an irreducible coherent family and \(\mu \in
+  \mathfrak{h}^*\). The following conditions are equivalent.
+  \begin{enumerate}
+    \item \(\mathcal{M}[\mu]\) is irreducible.
+    \item \(F_\alpha\!\restriction_{\mathcal{M}[\mu]}\) is injective for all
+      \(\alpha \in \Delta\).
+    \item \(\mathcal{M}[\mu]\) is cuspidal.
+  \end{enumerate}
+\end{theorem}

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -1859,7 +1859,7 @@ What is simultaneous diagonalization all about then?
   its elements commute with one another.
 \end{proposition}
 
-\begin{corollary}
+\begin{corollary}\label{thm:finite-dim-is-weight-mod}
   Let \(\mathfrak{g}\) be a Lie algebra, \(\mathfrak{h} \subset \mathfrak{g}\)
   be an Abelian subalgebra and \(V\) be any finite-dimensional representation
   of \(\mathfrak{g}\). Then there is a basis \(\{v_1, \ldots, v_n\}\) of \(V\)
@@ -2316,7 +2316,7 @@ is neither irreducible nor finite-dimensional. Nevertheless, we can use
 whose highest weight is \(\lambda\).
 
 % TODO: Adjust the notation for the maximal submodule
-\begin{proposition}
+\begin{proposition}\label{thm:max-verma-submod-is-weight}
   Every subrepresentation \(V \subset M(\lambda)\) is the direct sum of its
   weight spaces. In particular, \(M(\lambda)\) has a unique maximal
   subrepresentation \(N(\lambda)\) and a unique irreducible quotient

diff --git a/tcc.tex b/tcc.tex
@@ -24,6 +24,8 @@
 
 \input{sections/semisimple-algebras}
 
+\input{sections/mathieu}
+
 \printbibliography
 \end{document}