diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -343,12 +343,12 @@
% TODO: Remark that any module over the localization is a g-module if we
% restrict it via the localization map, wich is injective in this case
\begin{proposition}\label{thm:nice-automorphisms-exist}
- There is a family of automorphisms \(\{F_{\Sigma}^\lambda :
+ There is a family of automorphisms \(\{\theta_\lambda :
\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \to
\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\}_{\lambda \in
\mathfrak{h}^*}\) such that
\begin{enumerate}
- \item \(F_{\Sigma}^{k_1 \alpha_1 + \cdots k_n \alpha^n}(u) =
+ \item \(\theta_{k_1 \alpha_1 + \cdots k_n \alpha^n}(u) =
F_{\alpha_1}^{k_1} \cdots F_{\alpha_n}^{k_n} u F_{\alpha_1}^{- k_n}
\cdots F_{\alpha_n}^{- k_1}\) for all \(u \in
\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\) and \(k_1,
@@ -359,16 +359,16 @@
\begin{align*}
\mathfrak{h}^* &
\to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \\
- \lambda & \mapsto F_\Sigma^\lambda(u)
+ \lambda & \mapsto \theta_\lambda(u)
\end{align*}
is polynomial.
\item If \(M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
\Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
\Sigma)} \otimes V\), \(\lambda, \mu \in \mathfrak{h}^*\) and
- \(F_\Sigma^\lambda M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha :
+ \(\theta_\lambda M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha :
\alpha \in \Sigma)}\)-module \(M\) twisted by the automorphism
- \(F_\Sigma^\lambda\) then \(M_\mu = (F_\Sigma^\lambda M)_{\mu +
+ \(\theta_\lambda\) then \(M_\mu = (\theta_\lambda M)_{\mu +
\lambda}\).
\end{enumerate}
\end{proposition}
@@ -388,18 +388,18 @@
proposition~\ref{thm:irr-admissible-is-contained-in-nice-mod} and take
\[
\mathcal{M}
- = \bigoplus_{\lambda \in \Lambda} F_\Sigma^\lambda M
+ = \bigoplus_{\lambda \in \Lambda} \theta_\lambda M
\]
- On the one hand, \(V\) lies in \(M = F_\Sigma^0 M\) -- recall that
- \(F_\Sigma^0\) is just the identity operator -- and therefore \(V \subset
+ On the one hand, \(V\) lies in \(M = \theta_0 M\) -- recall that
+ \(\theta_0\) is just the identity operator -- and therefore \(V \subset
\mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
- F_\Sigma^\lambda M_\mu = \dim M_{\mu - \lambda} = d\) for all \(\mu \in
+ \theta_\lambda M_\mu = \dim M_{\mu - \lambda} = d\) for all \(\mu \in
\lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given \(u \in
C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) and \(\mu \in \lambda + Q\),
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
- = \operatorname{Tr}(F_\Sigma^\lambda(u)\!\restriction_{M_{\mu - \lambda}})
+ = \operatorname{Tr}(\theta_\lambda(u)\!\restriction_{M_{\mu - \lambda}})
\]
is polynomial in \(\mu\) because of the second item of
proposition~\ref{thm:nice-automorphisms-exist}.