lie-algebras-and-their-representations

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
   \(\lambda \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\)