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: fb51eaf69137727f891d20a41c64a1798a18e204
Parent: c20073ceafbc1d62e31e254d8471d97c9f45e385
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 14 Nov 2022 16:28:22 +0000

Moved the proposition on the multiplicity of irreducible weight modules inside completely reducible weight modules

Since this was a technical proposition, it was moved to right before it is first used

Diffstat

1 file changed, 23 insertions, 25 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	48	23	25

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -113,24 +113,6 @@ to the case it holds. This brings us to the following definition.
   \left(\mfrac{V}{W}\right)_\lambda\) is surjective.
 \end{example}
 
-% TODOO: Move this to somewhere else? This probably fits the best just before
-% or after the first result that uses it
-\begin{proposition}\label{thm:centralizer-multiplicity}
-  Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
-  \(V_\lambda\) is a semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any
-  \(\lambda \in \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is the
-  cetralizer\footnote{This notation comes from the fact that the centralizer of
-  $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ coincides the weight space
-  associated with $0 \in \mathfrak{h}^*$ in the adjoint action of
-  $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the
-  regular action of $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$.} of
-  \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\). Moreover, the multiplicity
-  of a given irreducible representation \(W\) of \(\mathfrak{g}\) coincides
-  with the multiplicity of \(W_\lambda\) in \(V_\lambda\) as a
-  \(\mathcal{U}(\mathfrak{g})_0\)-module, for any \(\lambda \in
-  \operatorname{supp} V\).
-\end{proposition}
-
 A particularly well behaved class of examples are the so called
 \emph{admissible} weight modules.
 
@@ -698,7 +680,23 @@ This last results provide a partial answer to the question of existence of nice
 coherent extensions. A complementary question now is: wich submodules of a nice
 coherent family are cuspidal representations?
 
-\begin{theorem}[Mathieu]
+\begin{proposition}\label{thm:centralizer-multiplicity}
+  Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
+  \(V_\lambda\) is a semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any
+  \(\lambda \in \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is the
+  cetralizer\footnote{This notation comes from the fact that the centralizer of
+  $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ coincides the weight space
+  associated with $0 \in \mathfrak{h}^*$ in the adjoint action of
+  $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the
+  regular action of $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$.} of
+  \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\). Moreover, the multiplicity
+  of a given irreducible representation \(W\) of \(\mathfrak{g}\) coincides
+  with the multiplicity of \(W_\lambda\) in \(V_\lambda\) as a
+  \(\mathcal{U}(\mathfrak{g})_0\)-module, for any \(\lambda \in
+  \operatorname{supp} V\).
+\end{proposition}
+
+\begin{corollary}[Mathieu]
   Let \(\mathcal{M}\) be a simple coherent family of degree \(d\) and
   \(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent.
   \begin{enumerate}
@@ -707,7 +705,7 @@ coherent family are cuspidal representations?
       all \(\alpha \in \Delta\).
     \item \(\mathcal{M}[\lambda]\) is cuspidal.
   \end{enumerate}
-\end{theorem}
+\end{corollary}
 
 \begin{proof}
   The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
@@ -735,12 +733,12 @@ coherent family are cuspidal representations?
   that \(\mathcal{M}_\mu\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module
   and \(\dim V_\mu = \deg V\).
 
-  % Here we use thm:centralizer-multiplicity
   In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any
-  irreducible \(\mathfrak{g}\)-module \(W\), the multiplicity of \(W\) in
-  \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
-  \(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module, which is, of
-  course, \(1\) if \(W \cong V\) and \(0\) otherwise. Hence
+  irreducible \(\mathfrak{g}\)-module \(W\), it follows from
+  proposition~\ref{thm:centralizer-multiplicity} that the multiplicity of \(W\)
+  in \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
+  \(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module -- which is,
+  of course, \(1\) if \(W \cong V\) and \(0\) otherwise. Hence
   \(\mathcal{M}[\lambda] = V\) and \(\mathcal{M}[\lambda]\) is cuspidal.
 \end{proof}