diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -64,12 +64,12 @@
\begin{proposition}\label{thm:centralizer-multiplicity}
Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
\(V_\lambda\) is a semisimple
- \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module for any \(\lambda \in
- \mathfrak{h}^*\), where \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) is
+ \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
+ \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is
the cetralizer of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\).
Moreover, the multiplicity of a given irreducible representation
\(W\) of \(\mathfrak{g}\) coincides with the multiplicity of \(W_\lambda\) in
- \(V_\lambda\) as a \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-module,
+ \(V_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module,
for any \(\lambda \in \operatorname{supp} V\).
\end{proposition}
@@ -167,7 +167,7 @@
\item \(\dim \mathcal{M}_\lambda = d\) for \emph{all} \(\lambda \in
\mathfrak{h}^*\)
\item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the centralizer
- \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) of \(\mathfrak{h}\) in
+ \(\mathcal{U}(\mathfrak{g})_0\) of \(\mathfrak{h}\) in
\(\mathcal{U}(\mathfrak{g})\), the map
\begin{align*}
\mathfrak{h}^* & \to K \\
@@ -187,7 +187,7 @@
\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 some \(\lambda \in
+ \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
\mathfrak{h}^*\).
\end{definition}
@@ -275,7 +275,7 @@
= d
\]
- Likewise, given \(u \in C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) the
+ Likewise, given \(u \in \mathcal{U}(\mathfrak{g})_0\) the
number
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}})
@@ -613,7 +613,7 @@
\subset \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
\theta_\lambda \Sigma^{-1} V_\mu = \dim \Sigma^{-1} V_{\mu - \lambda} = d\)
for all \(\mu \in \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given
- \(u \in C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) and \(\mu \in \lambda +
+ \(u \in \mathcal{U}(\mathfrak{g})_0\) and \(\mu \in \lambda +
Q\),
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
@@ -652,7 +652,7 @@
multiplicity of a given irreducible \(\mathfrak{g}\)-module \(W\) in
\(\mathcal{N}\) is determined by its \emph{trace function}
\begin{align*}
- \mathfrak{h}^* \times C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h}) &
+ \mathfrak{h}^* \times \mathcal{U}(\mathfrak{g})_0 &
\to K \\
(\lambda, u) &
\mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})
@@ -663,7 +663,7 @@
Indeed, given \(\lambda \in \operatorname{supp} V\) the multiplicity of \(W\)
in \(\mathcal{N}\) is the same as the multiplicity of \(W_\lambda\) in
\(\mathcal{N}_\lambda\), which is determined by the character
- \(\chi_{\mathcal{N}_\lambda} : C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})
+ \(\chi_{\mathcal{N}_\lambda} : \mathcal{U}(\mathfrak{g})_0
\to K\) -- see proposition~\ref{thm:centralizer-multiplicity}. We now claim
that the trace function of \(\mathcal{N}\) is the same as that of
\(\operatorname{Ext}(V)\). Clearly,
@@ -671,7 +671,7 @@
= \operatorname{Tr}(u\!\restriction_{V_\lambda})
= \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\) for all
\(\lambda \in \operatorname{supp}_{\operatorname{ess}} V\), \(u \in
- C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\). Since the essential support of
+ \mathcal{U}(\mathfrak{g})_0\). Since the essential support of
\(V\) is Zariski-dense and the maps \(\lambda \mapsto
\operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})\) and
\(\lambda \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\)