lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -253,48 +253,6 @@
   is polynomial in \(\mu \in \mathfrak{h}^*\).
 \end{proof}
 
-\begin{proposition}
-  Let \(V\) be a cuspidal representation of \(\mathfrak{g}\) and
-  \(\mathcal{M}\) be a semisimple coherent extension of \(V\) which is
-  irreducible as a coherent family. Then \(V = \mathcal{M}[\lambda]\) for any
-  \(\lambda \in \operatorname{supp} V\).
-\end{proposition}
-
-\begin{proof}
-  We know \(V\) is a subquotient of \(\mathcal{M}\). More precisely, since
-  \(V\) is irreducible it is a subquotient of \(\mathcal{M}[\lambda]\) -- its
-  support is contained in \(\lambda + Q\). Furthermore, once again it follows
-  from the irreducibility of \(V\) that it can be realized as the quotient of
-  consecutive terms of a composition series \(0 = \mathcal{M}_0 \subset
-  \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n = \mathcal{M}[\lambda]\).
-  But since \(\mathcal{M}\) is semisimple
-  \[
-    \mathcal{M}[\lambda]
-    \cong \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i},
-  \]
-  so that \(V\) is contained in \(\mathcal{M}[\lambda]\).
-
-  % TODOO: Here we need to take some care: there is some mu such that M_mu is
-  % simple, but this needs not to be the case
-  % TODOO: Our argument still works because we can apply it to such a mu and
-  % then argue that the dimension of the other weight spaces are maximal
-  % because of the fact that the E_alpha and F_alphas act injectively
-  Hence it suffices to show that \(V_\mu = \mathcal{M}_\mu\) for any
-  \(\mu \in \lambda + Q\). But this is already clear from the fact that
-  \(\mathcal{M}\) is irreducible as a coherent family: given \(v \in
-  V_\mu\), \(H \in \mathfrak{h}\) and \(u \in
-  C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) we find
-  \[
-    H u v = u H v = \mu(H) \cdot u v,
-  \]
-  so that \(V_\mu\) is a
-  \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-submodule of
-  \(\mathcal{M}_\mu\). 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
-  \(V_\mu = \mathcal{M}_\mu\).
-\end{proof}
-
 \begin{theorem}[Mathieu]
   Let \(\mathcal{M}\) be an irreducible coherent family and \(\lambda \in
   \mathfrak{h}^*\). The following conditions are equivalent.
@@ -378,7 +336,9 @@
   \(\mathfrak{g}\)-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
-  \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
+  \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\). Furthermore,
+  \(V\) is itself contained in \(\operatorname{Ext}(V)\) and
+  \(\operatorname{Ext}(V)\) is irreducible as a coherent family.
 \end{theorem}
 
 \begin{proof}
@@ -391,7 +351,7 @@
     = \bigoplus_{\lambda \in \Lambda} \theta_\lambda M
   \]
 
-  On the one hand, \(V\) lies in \(M = \theta_0 M\) -- recall that
+  On the one hand, \(V\) lies in \(M = \theta_0 M\) -- 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
@@ -405,30 +365,31 @@
   proposition~\ref{thm:nice-automorphisms-exist}.
 
   In other words, \(\mathcal{M}\) is a coherent extension of \(V\) of degree
-  \(d\). Hence there is a semisimple degree \(d\) coherent extention
-  \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of \(V\). To see
-  that \(\operatorname{Ext}(V)\) is irreducible, we note that \(V\) is
-  contained in \(\operatorname{Ext}(V)\). Indeed, if we fix some \(\lambda \in
-  \operatorname{supp} V\) and a composition series of \(0 = \mathcal{M}_0
-  \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
-  \mathcal{M}[\lambda]\) such that \(V \cong \mfrac{\mathcal{M}_{i +
-  1}}{\mathcal{M}_i}\) for some \(i\), there is a natural inclusion
+  \(d\). 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
   \[
     V
-    \isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
-    \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
+    \isoto \mfrac{\mathcal{M}_1}{\mathcal{M}_0}
+    \to \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
     = \mathcal{M}^{\operatorname{ss}}[\lambda]
     \subset \operatorname{Ext}(V)
   \]
 
   % TODOO: Prove that the weight spaces of any simple g-module are all simple
   % C(h)-modules
-  In particular, it follows from the irreducibility of \(V\) that
-  \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
-  \(C_{\mathcal{U}(\mathfrak{h})}(\mathfrak{h})\)-module for some \(\lambda \in
-  \operatorname{supp} V\): since the degree of \(V\) is the same as the
-  degree of \(\operatorname{Ext}(V)\) some of its weight spaces must have
-  maximal dimension inside of \(\operatorname{Ext}(V)\).
+  Since the degree of \(V\) is the same as the degree of
+  \(\operatorname{Ext}(V)\), some of its weight spaces have maximal dimension
+  inside of \(\operatorname{Ext}(V)\). In particular, it follows from the
+  irreducibility of \(V\) that \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is
+  a simple \(C_{\mathcal{U}(\mathfrak{h})}(\mathfrak{h})\)-module for some
+  \(\lambda \in \operatorname{supp} V\), so that \(\operatorname{Ext}(V)\) is
+  irreducible as a coherent family.
 
   % TODOO: Prove the uniqueness
 \end{proof}