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}