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 \(\lambda \in \mathfrak{h}^*\)