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
\[
V
- \isoto \mfrac{\mathcal{M}_1}{\mathcal{M}_0}
- \to \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+ \isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
+ \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
= \mathcal{M}^{\operatorname{ss}}[\lambda]
\subset \operatorname{Ext}(V)
\]