lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -21,7 +21,7 @@
   representation of a reductive Lie algebra is a weight module.
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:submod-is-weight-mod}
   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
@@ -31,7 +31,7 @@
   W\) for all \(\lambda \in \mathfrak{h}^*\).
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:quotient-is-weight-mod}
   Given a weight module \(V\), a submodule \(W \subset V\) and \(\lambda \in
   \mathfrak{h}^*\), \(\left(\mfrac{V}{W}\right)_\lambda = \mfrac{V_\lambda}{W}
   \cong \mfrac{V_\lambda}{W_\lambda}\). In particular,
@@ -173,25 +173,78 @@
 
 % Mathieu's proof of this is somewhat profane, I don't think it's worth
 % including it in here
+% TODO: Define the notation for M[mu] somewhere else
+% TODO: Note somewhere that M[mu] is a submodule
 \begin{proposition}
-  Any coherent family \(\mathcal{M}\) has finite length as a
-  \(\mathfrak{g}\)-module.
+  Given a coherent family \(\mathcal{M}\) and \(\lambda \in \mathfrak{h}^*\),
+  \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
 \end{proposition}
 
-% TODO: Add a proof!
+% TODO: Note that the semisimplification is only defined up to isomorphism: the
+% isomorphism class is independant of the composition series because all
+% composition series are conjugate
 \begin{corollary}
-  Given a coherent family \(\mathcal{M}\) and a composition series \(0 =
-  \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
-  \mathcal{M}\), the \(\mathfrak{g}\)-module
+  Let \(\{\lambda_i\}_i\) be a set of representatives of the \(Q\)-cosets of
+  \(\mathfrak{h}^*\).
+  Given a coherent family \(\mathcal{M}\) of degree \(d\) and composition
+  series \(0 = \mathcal{M}_0[\lambda_i] \subset \mathcal{M}_1[\lambda_i]
+  \subset \cdots \subset \mathcal{M}_n[\lambda_i] = \mathcal{M}[\lambda_i]\),
+  the \(\mathfrak{g}\)-module
   \[
     \mathcal{M}^{\operatorname{ss}}
-    = \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+    = \bigoplus_{i j}
+      \mfrac{\mathcal{M}_{j + 1}[\lambda_i]}{\mathcal{M}_j[\lambda_i]}
   \]
-  is also a coherent family, called \emph{the semisimplification\footnote{This
-  name is due to the fact that $\mathcal{M}^{\operatorname{ss}}$ is the direct
-  sum of irreducible $\mathfrak{g}$-modules} of \(\mathcal{M}\)}.
+  is also a coherent family of degree \(d\), called \emph{the
+  semisimplification\footnote{This name is due to the fact that
+  $\mathcal{M}^{\operatorname{ss}}$ is the direct sum of irreducible
+  $\mathfrak{g}$-modules.} of \(\mathcal{M}\)}.
 \end{corollary}
 
+\begin{proof}
+  We know form examples~\ref{ex:submod-is-weight-mod} and
+  \ref{ex:quotient-is-weight-mod} that each quotient \(\mfrac{\mathcal{M}_{j +
+  1}[\lambda_i]}{\mathcal{M}_j[\lambda_i]}\) is weight module. Hence
+  \(\mathcal{M}^{\operatorname{ss}}\) is a weight module. Furtheremore, given
+  \(\mu \in \mathfrak{h}^*\)
+  \[
+    \mathcal{M}_\mu^{\operatorname{ss}}
+    = \bigoplus_{i j}
+      \left(
+      \mfrac{\mathcal{M}_{j + 1}[\lambda_i]}{\mathcal{M}_j[\lambda_i]}
+      \right)_\mu
+    \cong \bigoplus_{i j}
+      \mfrac{\mathcal{M}_{j + 1}[\lambda_i]_\mu}
+            {\mathcal{M}_j[\lambda_i]_\mu}
+  \]
+
+  In particular,
+  \[
+    \dim \mathcal{M}_\mu^{\operatorname{ss}}
+    = \sum_{i j}
+      \dim \mathcal{M}_{j + 1}[\lambda_i]_\mu
+    - \dim \mathcal{M}_j[\lambda_i]_\mu
+    = \sum_i \dim \mathcal{M}[\lambda_i]_\mu
+    = \dim \mathcal{M}_\mu
+    = d
+  \]
+
+  Likewise, given \(u \in C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) the
+  number
+  \[
+    \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}})
+    = \sum_{i j}
+      \operatorname{Tr}
+      (u\!\restriction_{\mathcal{M}_{j + 1}[\lambda_i]_\mu})
+    - \operatorname{Tr}
+      (u\!\restriction_{\mathcal{M}_j[\lambda_i]_\mu})
+    = \sum_{i}
+      \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\lambda_i]_\mu})
+    = \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
+  \]
+  is polynomial in \(\mu \in \mathfrak{h}^*\).
+\end{proof}
+
 \begin{theorem}[Mathieu]
   Let \(V\) be an infinite-dimensional admissible irreducible
   \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
@@ -201,7 +254,7 @@
   \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
 \end{theorem}
 
-% TODO: Define the notation for M[mu] somewhere else
+% TODOO: This is already noted in a previous corollary by Fernando
 % TODO: Note somewhere else that the support of a cuspidal module is an entire
 % Q-coset
 \begin{proposition}