lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1334,28 +1334,28 @@ Lo and behold\dots
 
 \begin{theorem}[Mathieu]
   There exists a unique completely reducible coherent extension
-  \(\operatorname{Ext}(V)\) of \(V\). More precisely, if \(\mathcal{M}\) is any
+  \(\mathcal{Ext}(V)\) of \(V\). More precisely, if \(\mathcal{M}\) is any
   coherent extension of \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong
-  \operatorname{Ext}(V)\). Furthermore, \(\operatorname{Ext}(V)\) is
+  \mathcal{Ext}(V)\). Furthermore, \(\mathcal{Ext}(V)\) is
   a simple coherent family.
 \end{theorem}
 
 \begin{proof}
   The existence part should be clear from the previous discussion: it suffices
   to fix some coherent extension \(\mathcal{M}\) of \(V\) and take
-  \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
+  \(\mathcal{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
 
-  To see that \(\operatorname{Ext}(V)\) is simple, recall from
+  To see that \(\mathcal{Ext}(V)\) is simple, recall from
   Corollary~\ref{thm:admissible-is-submod-of-extension} that \(V\) is a
-  subrepresentation 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
+  subrepresentation of \(\mathcal{Ext}(V)\). Since the degree of \(V\) is
+  the same as the degree of \(\mathcal{Ext}(V)\), some of its weight
+  spaces have maximal dimension inside of \(\mathcal{Ext}(V)\). In
   particular, it follows from Proposition~\ref{thm:centralizer-multiplicity}
-  that \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
+  that \(\mathcal{Ext}(V)_\lambda = V_\lambda\) is a simple
   \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
   \operatorname{supp} V\).
 
-  As for the uniqueness of \(\operatorname{Ext}(V)\), fix some other completely
+  As for the uniqueness of \(\mathcal{Ext}(V)\), fix some other completely
   reducible coherent extension \(\mathcal{N}\) of \(V\). We claim that the
   multiplicity of a given irreducible \(\mathfrak{g}\)-module \(W\) in
   \(\mathcal{N}\) is determined by its \emph{trace function}
@@ -1380,21 +1380,21 @@ Lo and behold\dots
   \mathcal{U}(\mathfrak{g})_0 \to K\). Since this holds for all irreducible
   weight \(\mathfrak{g}\)-modules, it follows that \(\mathcal{N}\) is
   determined by its trace function. Of course, the same holds for
-  \(\operatorname{Ext}(V)\). We now claim that the trace function of
-  \(\mathcal{N}\) is the same as that of \(\operatorname{Ext}(V)\). Clearly,
-  \(\operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda}) =
+  \(\mathcal{Ext}(V)\). We now claim that the trace function of
+  \(\mathcal{N}\) is the same as that of \(\mathcal{Ext}(V)\). Clearly,
+  \(\operatorname{Tr}(u\!\restriction_{\mathcal{Ext}(V)_\lambda}) =
   \operatorname{Tr}(u\!\restriction_{V_\lambda}) =
   \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\) for all \(\lambda
   \in \operatorname{supp}_{\operatorname{ess}} V\), \(u \in
   \mathcal{U}(\mathfrak{g})_0\). Since the essential support of \(V\) is
   Zariski-dense and the maps \(\lambda \mapsto
-  \operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})\) and
+  \operatorname{Tr}(u\!\restriction_{\mathcal{Ext}(V)_\lambda})\) and
   \(\lambda \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\)
   are polynomial in \(\lambda \in \mathfrak{h}^*\), it follows that these maps
   coincide for all \(u\).
 
-  In conclusion, \(\mathcal{N} \cong \operatorname{Ext}(V)\) and
-  \(\operatorname{Ext}(V)\) is unique.
+  In conclusion, \(\mathcal{N} \cong \mathcal{Ext}(V)\) and
+  \(\mathcal{Ext}(V)\) is unique.
 \end{proof}
 
 % This is a very important theorem, but since we won't classify the coherent