diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -8,12 +8,6 @@
\(\operatorname{supp} V = \{\lambda \in \mathfrak{h}^* : V_\lambda \ne 0\}\).
\end{definition}
-\begin{definition}
- A weight \(\mathfrak{g}\)-module is called \emph{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 representation of a semisimple Lie algebra is a
@@ -66,6 +60,35 @@
% TODO: Add an example of a module wich is NOT a weight module
+% TODOO: Prove this?
+\begin{proposition}\label{thm:centralizer-multiplicity}
+ Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
+ \(V_\lambda\) is a semisimple
+ \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module for any \(\lambda \in
+ \mathfrak{h}^*\), where \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) is
+ the cetralizer 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 \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module,
+ for any \(\lambda \in \operatorname{supp} V\).
+\end{proposition}
+
+\begin{definition}
+ A weight \(\mathfrak{g}\)-module is called \emph{admissible} if \(\dim
+ V_\lambda\) is bounded. The lowest upper bound for \(\dim V_\lambda\) is
+ called \emph{the degree of \(V\)}. The \emph{essential support} of \(V\) is
+ the set \(\operatorname{supp}_{\operatorname{ess}} V = \{ \lambda \in
+ \mathfrak{h}^* : \dim V_\lambda = d \}\).
+\end{definition}
+
+% This proof is very technical, I don't think its worth including it
+\begin{proposition}
+ Let \(V\) be an infinite-dimensional admissible representation of
+ \(\mathfrak{g}\). The essential support
+ \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense in
+ \(\mathfrak{h}^*\).
+\end{proposition}
+
\begin{definition}
A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
if \(\mathfrak{b} \subset \mathfrak{p}\).
@@ -206,7 +229,6 @@
\end{corollary}
\begin{proof}
- % TODOOO: Note that any submodule of a semisimple module is semisimple
The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear:
since \(\mathcal{M}^{\operatorname{ss}}\) is completely reducible, so is
\(\mathcal{M}^{\operatorname{ss}}[\lambda_i]\). Hence
@@ -610,23 +632,54 @@
irreducible as a coherent family.
\end{theorem}
-% TODOOO: Prove the uniqueness!
\begin{proof}
The existence part should be clear from the previous discussion: it suffices
to fix some coherent extension \(\mathcal{M}\) of \(V\) and take
\(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
- % TODOOO: Prove that the weight spaces of any simple g-module are all simple
- % C(h)-modules
To see that \(\operatorname{Ext}(V)\) is irreducible as a coherent family,
recall from corollary~\ref{thm:admissible-is-submod-of-extension} that \(V\)
is a subrepresentation of \(\operatorname{Ext}(V)\). Since the degree of
\(V\) is the same as the degree of \(\operatorname{Ext}(V)\), some of its
weight spaces have maximal dimension inside of \(\operatorname{Ext}(V)\). In
- particular, it follows from the irreducibility of \(V\) that
- \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
+ particular, it follows from proposition~\ref{thm:centralizer-multiplicity}
+ that \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
\(C_{\mathcal{U}(\mathfrak{h})}(\mathfrak{h})\)-module for some \(\lambda \in
\operatorname{supp} V\).
+
+ As for the uniqueness of \(\operatorname{Ext}(V)\), fix some other completely
+ reducible coherent extension \(\mathcal{N}\) of \(V\). We claim that the
+ multiplicity of a given irreducible \(\mathfrak{g}\)-module \(W\) in
+ \(\mathcal{N}\) is determined by its \emph{trace function}
+ \begin{align*}
+ \mathfrak{h}^* \times C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h}) &
+ \to K \\
+ (\lambda, u) &
+ \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})
+ \end{align*}
+
+ % TODO: Point out that this multiplicity is determined by the characters
+ % beforehand
+ Indeed, given \(\lambda \in \operatorname{supp} V\) the multiplicity of \(W\)
+ in \(\mathcal{N}\) is the same as the multiplicity of \(W_\lambda\) in
+ \(\mathcal{N}_\lambda\), which is determined by the character
+ \(\chi_{\mathcal{N}_\lambda} : C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})
+ \to K\) -- see proposition~\ref{thm:centralizer-multiplicity}. We now claim
+ that the trace function of \(\mathcal{N}\) is the same as that of
+ \(\operatorname{Ext}(V)\). Clearly,
+ \(\operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})
+ = \operatorname{Tr}(u\!\restriction_{V_\lambda})
+ = \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\) for all
+ \(\lambda \in \operatorname{supp}_{\operatorname{ess}} V\), \(u \in
+ C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\). Since the essential support of
+ \(V\) is Zariski-dense and the maps \(\lambda \mapsto
+ \operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})\) and
+ \(\lambda \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\)
+ are polynomial in \(\lambda \in \mathfrak{h}^*\), it follows that this maps
+ coincide for all \(u\).
+
+ In conclusion, \(\mathcal{N} \cong \operatorname{Ext}(V)\) and
+ \(\operatorname{Ext}(V)\) is unique.
\end{proof}
\begin{proposition}[Mathieu]