lie-algebras-and-their-representations

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]