diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -120,21 +120,42 @@
which is \emph{not} parabolic induced.
\end{definition}
-% TODOOO: w should be an element of the subgroup W(V) of W. Fix this!
% TODO: Remark on the fact that any simple weight p-mod is a (p/u)-mod, so that
% the notation of a cuspidal p-mod is well definited
-% TODO: Define the conjugation of a p-mod by an element of the Weil group
+% TODO: Define the conjugation of a p-mod by an element of the Weil group?
\begin{theorem}[Fernando]
Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to
\(L_{\mathfrak{p}}(V)\) for some parabolic subalgebra \(\mathfrak{p} \subset
\mathfrak{g}\) and some irreducible cuspidal \(\mathfrak{p}\)-module \(V\).
- Furthermore, if \(\mathfrak{p}_1, \mathfrak{p}_2 \subset \mathfrak{g}\) are
- parabolic and \(V_i\) is an irreducible cuspidal \(\mathfrak{p}_i\)-module
- then \(L_{\mathfrak{p}_1}(V_1) \cong L_{\mathfrak{p}_2}(V_2)\) if, and only
- if \(\mathfrak{p}_1 = \mathfrak{p}_2^w\) and \(V_1 \cong V_2^w\) for some \(w
- \in \mathcal{W}\).
\end{theorem}
+% TODO: Point out that the relationship between p-modules and cuspidal
+% g-modules is not 1-to-1
+
+\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 \(\Sigma \subset
+ \Delta_{\mathfrak{p}_1}\). 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 \(\mathfrak{p}' = \mathfrak{p}^w\) and
+ \(W \cong V^w\) for some \(w \in \mathcal{W}_V\), where
+ \[
+ \mathcal{W}_V
+ = \langle
+ T_\beta : \beta \in \Sigma, H_\beta + \mathfrak{u}
+ \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{u}}
+ \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ V
+ \rangle
+ \subset \mathcal{W}
+ \]
+\end{proposition}
+
+% TODO: Point out that the definition of W_V is independant of the choice of
+% Sigma
+
% TODO: Remark that the support of a simple weight module is always contained
% in a coset
% TODO: Note that conditions (ii), (iii) and (iv) have special names