lie-algebras-and-their-representations

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]