lie-algebras-and-their-representations

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}.