diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -228,8 +228,7 @@ construction very similar to that of Verma modules.
\begin{definition}
The module \(M_{\mathfrak{p}}(V) =
\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is called \emph{a
- generalized Verma module}\footnote{It should be clear from the definitions
- that Verma modules are indeed generalized Verma modules.}.
+ generalized Verma module}.
\end{definition}
\begin{proposition}\label{thm:generalized-verma-has-simple-quotient}
@@ -273,19 +272,21 @@ We should point out that the relationship between irreducible weight
cuspidal \(\mathfrak{p}\)-module -- is not one-to-one. Nevertheless, this
relationship is well understood. Namely, Fernando himself established\dots
-% TODOO: Define the conjugation of a p-mod by an element of the Weil group
-% beforehand
\begin{proposition}[Fernando]
Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\), there
- exists a basis\footnote{This is usually called \emph{a $\mathfrak{p}$-adapted
- basis.}} \(\Sigma\) for \(\Delta\) such that \(\Sigma \subset
+ exists a basis \(\Sigma\) for \(\Delta\) such that \(\Sigma \subset
\Delta_{\mathfrak{p}'}\). Furthermore, if \(\mathfrak{p}' \subset
\mathfrak{g}\) is another parabolic subalgebra, \(V\) is an irreducible
cuspidal \(\mathfrak{p}\)-module and \(W\) is an irreducible cuspidal
\(\mathfrak{p}'\)-module then \(L_{\mathfrak{p}}(V) \cong
L_{\mathfrak{p}'}(W)\) if, and only if \(\mathfrak{p}' =
- \mathfrak{p}^\sigma\) and \(W \cong V^\sigma\) for some \(\sigma \in
- \mathcal{W}_V\), where
+ \mathfrak{p}^\sigma\) and \(V \cong \sigma W\) for some\footnote{Here
+ $\mathfrak{p}^\sigma$ denotes the image of $\mathfrak{p}$ under the
+ automorphism of $\sigma : \mathfrak{g} \to \mathfrak{g}$ given by the
+ canonical action of $\mathcal{W}$ in $\mathfrak{g}$ and $\sigma W$ is the
+ $\mathfrak{p}$-module given by composing the map $\mathfrak{p}' \to
+ \mathfrak{gl}(W)$ with the restriction $\sigma\!\restriction_{\mathfrak{p}} :
+ \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in \mathcal{W}_V\), where
\[
\mathcal{W}_V
= \langle