lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -163,7 +163,7 @@
 \begin{definition}
   A coherent family \(\mathcal{M}\) called \emph{irreducible} if
   \(\mathcal{M}_\lambda\) is a simple
-  \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module for each \(\lambda \in
+  \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module for some \(\lambda \in
   \mathfrak{h}^*\).
 \end{definition}
 
@@ -274,6 +274,11 @@
   \]
   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
@@ -328,7 +333,7 @@
   is injective.
 \end{corollary}
 
-\begin{proposition}
+\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 \(\operatorname{supp} M = Q + \operatorname{supp} V\)
@@ -337,7 +342,7 @@
 
 % TODO: Remark that any module over the localization is a g-module if we
 % restrict it via the localization map, wich is injective in this case
-\begin{proposition}
+\begin{proposition}\label{thm:nice-automorphisms-exist}
   There is a family of automorphisms \(\{F_{\Sigma}^\lambda :
   \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \to
   \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\}_{\lambda \in
@@ -376,6 +381,58 @@
   \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
 \end{theorem}
 
+\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
+  proposition~\ref{thm:irr-admissible-is-contained-in-nice-mod} and take
+  \[
+    \mathcal{M}
+    = \bigoplus_{\lambda \in \Lambda} F_\Sigma^\lambda M
+  \]
+
+  On the one hand, \(V\) lies in \(M = F_\Sigma^0 M\) -- recall that
+  \(F_\Sigma^0\) is just the identity operator -- and therefore \(V \subset
+  \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
+  F_\Sigma^\lambda M_\mu = \dim M_{\mu - \lambda} = d\) for all \(\mu \in
+  \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given \(u \in
+  C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) and \(\mu \in \lambda + Q\),
+  \[
+    \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
+    = \operatorname{Tr}(F_\Sigma^\lambda(u)\!\restriction_{M_{\mu - \lambda}})
+  \]
+  is polynomial in \(\mu\) because of the second item of
+  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
+  \[
+    V
+    \isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+    \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
+    = \mathcal{M}^{\operatorname{ss}}[\lambda]
+    \subset \operatorname{Ext}(V)
+  \]
+
+  % 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 \(\operatorname{Ext}(V)\).
+
+  % TODOO: Prove the uniqueness
+\end{proof}
+
 \begin{proposition}[Mathieu]
   The central characters of the irreducible submodules of
   \(\operatorname{Ext}(V)\) are all the same.