lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -460,18 +460,19 @@
   is a cuspidal representation, and its degree is bounded by \(d\). We claim
   \(\mathcal{M}[\lambda] = V\).
 
-  Since \(U = \{\mu \in \mathfrak{h}^* : \mathcal{M}_\mu \ \text{is a
-  simple $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is a non-empty --
-  \(\mathcal{M}\) is irreducible -- open set and
-  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense, \(U \cap
-  \operatorname{supp}_{\operatorname{ess}} V\) is non-empty. In other words,
-  there is some \(\mu \in \mathfrak{h}^*\) such that \(\mathcal{M}_\mu\) is a
-  simple \(\mathcal{U}(\mathfrak{g})_0\)-module and \(\dim V_\mu = \deg V\).
+  Since \(\mathcal{M}\) is irreducible and
+  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense, \(U = \{\mu
+  \in \mathfrak{h}^* : \mathcal{M}_\mu \ \text{is a simple
+  $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is a non-empty open set, and \(U
+  \cap \operatorname{supp}_{\operatorname{ess}} V\) is non-empty. In other
+  words, there is some \(\mu \in \mathfrak{h}^*\) such that \(\mathcal{M}_\mu\)
+  is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module and \(\dim V_\mu = \deg
+  V\).
 
   In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any
   irreducible \(\mathfrak{g}\)-module \(W\), the multiplicity of \(W\) in
   \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
-  \(\mathcal{M}\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module, which is, of
+  \(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module, which is, of
   course, \(1\) if \(W \cong V\) and \(0\) otherwise. Hence
   \(\mathcal{M}[\lambda] = V\) and \(\mathcal{M}[\lambda]\) is cuspidal.
 \end{proof}
@@ -726,13 +727,11 @@
 
   But
   \[
-    \begin{split}
-      \theta_\beta(H)
-      & = F_\beta H F_\beta^{-1} \\
-      & = ([F_\beta, H] + H F_\beta) F_\beta^{-1} \\
-      & = (\beta(H) + H) F_\beta F_\beta^{-1} \\
-      & = \beta(H) + H
-    \end{split}
+    \theta_\beta(H)
+    = F_\beta H F_\beta^{-1}
+    = ([F_\beta, H] + H F_\beta) F_\beta^{-1}
+    = (\beta(H) + H) F_\beta F_\beta^{-1}
+    = \beta(H) + H
   \]
   for all \(H \in \mathfrak{h}\) and \(\beta \in \Sigma\). In general,
   \(\theta_\lambda(H) = \lambda(H) + H\) for all \(\lambda \in \mathfrak{h}^*\)