diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -273,20 +273,20 @@
% TODO: Define what a set commuting roots is
\begin{lemma}\label{thm:nice-basis-for-inversion}
Let \(V\) be an irreducible infinite-dimensional admissible
- \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\alpha_1, \ldots,
- \alpha_n\}\) of \(Q\) consisting a commuting roots and such that the elements
- \(F_{\alpha_i}\) all act injectively on \(V\).
+ \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
+ \beta_n\}\) of \(Q\) consisting a commuting roots and such that the elements
+ \(F_{\beta_i}\) all act injectively on \(V\).
\end{lemma}
\begin{corollary}
Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
- \((F_\alpha : \alpha \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
- the multiplicative subset generated by \(F_\alpha\), \(\alpha \in \Sigma\).
- The \(K\)-algebra \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+ \((F_\beta : \beta \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
+ the multiplicative subset generated by \(F_\beta\), \(\beta \in \Sigma\).
+ The \(K\)-algebra \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
\Sigma)}\) is well defined and the localization map
\begin{align*}
V &
- \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}
+ \to \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
\otimes V \\
u & \mapsto 1 \otimes u
\end{align*}
@@ -304,30 +304,30 @@
% 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 \(\{\theta_\lambda :
- \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \to
- \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\}_{\lambda \in
+ \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)} \to
+ \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\}_{\lambda \in
\mathfrak{h}^*}\) such that
\begin{enumerate}
- \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,
+ \item \(\theta_{k_1 \beta_1 + \cdots k_n \beta_n}(u) =
+ F_{\beta_1}^{k_1} \cdots F_{\beta_n}^{k_n} u F_{\beta_1}^{- k_n}
+ \cdots F_{\beta_n}^{- k_1}\) for all \(u \in
+ \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\) and \(k_1,
\ldots, k_n \in \mathbb{Z}\).
- \item For each \(u \in \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+ \item For each \(u \in \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
\Sigma)}\) the map
\begin{align*}
\mathfrak{h}^* &
- \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \\
+ \to \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)} \\
\lambda & \mapsto \theta_\lambda(u)
\end{align*}
is polynomial.
- \item If \(W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
- \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+ \item If \(W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
+ \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
\Sigma)} \otimes V\), \(\lambda, \mu \in \mathfrak{h}^*\) and
- \(\theta_\lambda W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha :
- \alpha \in \Sigma)}\)-module \(W\) twisted by the automorphism
+ \(\theta_\lambda W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\beta :
+ \beta \in \Sigma)}\)-module \(W\) twisted by the automorphism
\(\theta_\lambda\) then \(W_\mu = (\theta_\lambda W)_{\mu +
\lambda}\).
\end{enumerate}
@@ -342,7 +342,7 @@
\begin{proof}
Let \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
\(\mathfrak{h}^*\) with \(0 \in \Lambda\), \(W =
- \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}
+ \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
\otimes_{\mathcal{U}(\mathfrak{g})} V\) be as in
proposition~\ref{thm:irr-admissible-is-contained-in-nice-mod} and take
\[