lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -295,9 +295,9 @@
 
 \begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
   Let \(V\) be an irreducible infinite-dimensional admissible
-  \(\mathfrak{g}\)-module. Then \(V\) is contained in a weight module \(M\) of
-  degree \(d\) such that \(\operatorname{supp} M = Q + \operatorname{supp} V\)
-  and \(\dim M_\lambda = d\) for all \(\lambda \in \operatorname{supp} M\).
+  \(\mathfrak{g}\)-module. Then \(V\) is contained in a weight module \(W\) of
+  degree \(d\) such that \(\operatorname{supp} W = Q + \operatorname{supp} V\)
+  and \(\dim W_\lambda = d\) for all \(\lambda \in \operatorname{supp} W\).
 \end{proposition}
 
 % TODO: Remark that any module over the localization is a g-module if we
@@ -323,63 +323,75 @@
       \end{align*}
       is polynomial.
 
-    \item If \(M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+    \item If \(W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
       \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
       \Sigma)} \otimes V\), \(\lambda, \mu \in \mathfrak{h}^*\) and
-      \(\theta_\lambda M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha :
-      \alpha \in \Sigma)}\)-module \(M\) twisted by the automorphism
-      \(\theta_\lambda\) then \(M_\mu = (\theta_\lambda M)_{\mu +
+      \(\theta_\lambda W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha :
+      \alpha \in \Sigma)}\)-module \(W\) twisted by the automorphism
+      \(\theta_\lambda\) then \(W_\mu = (\theta_\lambda W)_{\mu +
       \lambda}\).
   \end{enumerate}
 \end{proposition}
 
-\begin{theorem}[Mathieu]
+\begin{proposition}[Mathieu]
   Let \(V\) be an infinite-dimensional admissible irreducible
-  \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
-  coherent extension \(\operatorname{Ext}(V)\) of \(V\). More precisely, if
-  \(\mathcal{M}\) is any coherent extension of \(V\), then
-  \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\). Furthermore,
-  \(V\) is itself contained in \(\operatorname{Ext}(V)\) and
-  \(\operatorname{Ext}(V)\) is irreducible as a coherent family.
-\end{theorem}
+  \(\mathfrak{g}\)-module of degree \(d\). There exists a coherent extension
+  \(\mathcal{M}\) of \(V\) with degree \(d\).
+\end{proposition}
 
-% TODOOO: Prove the uniqueness
 \begin{proof}
-  The existence part should now be clear from the previous discussion: let
-  \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
-  \(\mathfrak{h}^*\) with \(0 \in \Lambda\), \(M\) be as in
+  Let \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
+  \(\mathfrak{h}^*\) with \(0 \in \Lambda\), \(W =
+  \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}
+  \otimes_{\mathcal{U}(\mathfrak{g})} V\) be as in
   proposition~\ref{thm:irr-admissible-is-contained-in-nice-mod} and take
   \[
     \mathcal{M}
-    = \bigoplus_{\lambda \in \Lambda} \theta_\lambda M
+    = \bigoplus_{\lambda \in \Lambda} \theta_\lambda W
   \]
 
-  On the one hand, \(V\) lies in \(M = \theta_0 M\) -- notice that
+  On the one hand, \(V\) lies in \(W = \theta_0 W\) -- notice that
   \(\theta_0\) is just the identity operator -- and therefore \(V \subset
   \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
-  \theta_\lambda M_\mu = \dim M_{\mu - \lambda} = d\) for all \(\mu \in
+  \theta_\lambda W_\mu = \dim W_{\mu - \lambda} = d\) for all \(\mu \in
   \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given \(u \in
   C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) and \(\mu \in \lambda + Q\),
   \[
     \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
-    = \operatorname{Tr}(\theta_\lambda(u)\!\restriction_{M_{\mu - \lambda}})
+    = \operatorname{Tr}(\theta_\lambda(u)\!\restriction_{W_{\mu - \lambda}})
   \]
   is polynomial in \(\mu\) because of the second item of
   proposition~\ref{thm:nice-automorphisms-exist}.
+\end{proof}
 
-  In other words, \(\mathcal{M}\) is a coherent extension of \(V\) of degree
-  \(d\). Hence there is a semisimple coherent extention \(\operatorname{Ext}(V)
-  = \mathcal{M}^{\operatorname{ss}}\) of \(V\) with \(\deg
-  \operatorname{Ext}(V) = d\). We claim \(V\) is contained in
-  \(\operatorname{Ext}(V)\). Indeed, since \(V\) is contained in \(M \subset
-  \mathcal{M}\) given \(\lambda \in \operatorname{supp} V\) we can fix some
-  composition series of \(0 = \mathcal{M}_0 \subset \mathcal{M}_1 \subset
-  \cdots \subset \mathcal{M}_n = \mathcal{M}[\lambda]\) with that \(V =
-  \mathcal{M}_1\), so that there is a natural inclusion
+\begin{theorem}[Mathieu]
+  Let \(V\) be an infinite-dimensional admissible irreducible
+  \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
+  coherent extension \(\operatorname{Ext}(V)\) of \(V\). More precisely, if
+  \(\mathcal{M}\) is any coherent extension of \(V\), then
+  \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\). Furthermore,
+  \(V\) is itself contained in \(\operatorname{Ext}(V)\) and
+  \(\operatorname{Ext}(V)\) is irreducible as a coherent family.
+\end{theorem}
+
+% TODOOO: Prove the uniqueness
+% TODOO: Does every coherent extension of V have the same degree as V?
+\begin{proof}
+  The existence part should be clear from the previous discussion: let
+  \(\mathcal{M}\) be a coherent extension of \(V\) with \(\deg \mathcal{M} =
+  d\) and take \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of
+  \(V\). 
+
+  We claim \(V\) is contained in \(\operatorname{Ext}(V)\). Indeed, if we fix
+  some composition series \(0 =
+  \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
+  \mathcal{M}[\lambda]\) of \(\mathcal{M}[\lambda]\) with
+  \(V \cong \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}\), \(\lambda \in
+  \operatorname{supp} V\), there is a natural inclusion
   \[
     V
-    \isoto \mfrac{\mathcal{M}_1}{\mathcal{M}_0}
-    \to \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+    \isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+    \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
     = \mathcal{M}^{\operatorname{ss}}[\lambda]
     \subset \operatorname{Ext}(V)
   \]