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}