diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -114,7 +114,7 @@
% TODO: Remark that the support of a simple weight module is always contained
% in a coset
% TODO: Note that conditions (ii) and (iii) have special names
-\begin{corollary}[Fernando]
+\begin{corollary}[Fernando]\label{thm:cuspidal-mod-equivs}
Let \(V\) be an irreducible weight \(\mathfrak{g}\)-module. The following
conditions are equivalent.
\begin{enumerate}
@@ -256,9 +256,6 @@
\(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
\end{theorem}
-% 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}
Let \(V\) be a cuspidal representation of \(\mathfrak{g}\) and take any
weight \(\lambda\) of \(V\). Then \(V \cong
@@ -290,11 +287,10 @@
\]
so that \(V_\mu\) is a
\(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-submodule of
- \(\operatorname{Ext}(V)_\mu\).
- Since \(V\) is cuspidal, \(\mu \in \lambda + Q = \operatorname{supp} V\)
- implies \(V_\mu \ne 0\), and hence \(V_\mu = \operatorname{Ext}(V)_\mu\) --
- because \(\operatorname{Ext}(V)_\mu\) is an irreducible
- \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module.
+ \(\operatorname{Ext}(V)_\mu\). Since \(V\) is cuspidal, \(\mu \in \lambda + Q
+ = \operatorname{supp} V\) -- i.e. the third equivalence of
+ corollary~\ref{thm:cuspidal-mod-equivs} -- implies \(V_\mu \ne 0\), and hence
+ \(V_\mu = \operatorname{Ext}(V)_\mu\).
\end{proof}
\begin{theorem}[Mathieu]