lie-algebras-and-their-representations

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

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -566,8 +566,47 @@ class of arguments leads us to the conclusion\dots
   \(\mathcal{W}\).
 \end{theorem}
 
-Now the only thing we are missing for a complete classification is an existence
-and uniqueness theorem analogous to theorem~\ref{thm:sl2-exist-unique} and
+Aside from showing up the previous theorem, the Weyl group will also play an
+important role in chapter~\ref{ch:mathieu} by virtue of the existance of a
+canonical action of \(\mathcal{W}\) in \(\mathfrak{h}\). By its very nature
+\(\mathcal{W}\) acts in \(\mathfrak{h}^*\). If we conjugate the action
+\(\sigma\!\restriction_{\mathfrak{h}^*} : \mathfrak{h}^* \isoto
+\mathfrak{h}^*\) of some \(\sigma \in \mathcal{W}\) by the isomorphism
+\(\mathfrak{h}^* \isoto \mathfrak{h}\) afforded by the restriction of the
+Killing for to \(\mathfrak{h}\) we get a linear automorphism \(\mathfrak{h}
+\isoto \mathfrak{h}\). As it turns out, the automorphism
+\(\sigma\!\restriction_{\mathfrak{h}} : \mathfrak{h} \isoto \mathfrak{h}\) can
+be extended to an automorphism of Lie algebras \(\mathfrak{g} \isoto
+\mathfrak{g}\). Namely\dots
+
+\begin{proposition}\label{thm:weyl-group-action}
+  Given \(\alpha \in \Delta^+\), let\footnote{Notice that since $\mathfrak{g}$
+  is finite-dimensional, $\operatorname{ad}(X)$ is nilpotent for each root
+  vector $X \in \mathfrak{g}$, so that the linear automorphism
+  $e^{\operatorname{ad}(X)} = \operatorname{Id} + \operatorname{ad}(X) +
+  \frac{\operatorname{ad}(X)^2}{2!} + \cdots : \mathfrak{g} \isoto
+  \mathfrak{g}$ is well defined.} \(\tilde{\sigma}_\alpha =
+  e^{\operatorname{ad}(E_\alpha)} e^{- \operatorname{ad}(F_\alpha)}
+  e^{\operatorname{ad}(E_\alpha)} : \mathfrak{g} \isoto \mathfrak{g}\). Then
+  \(\tilde\sigma_\alpha\) is an automorphism of Lie algebras, and this defines
+  an action of \(\mathcal{W}\) in \(\mathfrak{g}\) which is compatible with the
+  canonical action of \(\mathcal{W}\) in \(\mathfrak{h}\) -- i.e.
+  \(\tilde\sigma\!\restriction_{\mathfrak{h}} =
+  \sigma\!\restriction_{\mathfrak{h}}\) for all \(\sigma \in \mathcal{W}\).
+\end{proposition}
+
+\begin{note}
+  Notice that the action of \(\mathcal{W}\) in \(\mathfrak{g}\) from
+  proposition~\ref{thm:weyl-group-action} is not canonical, since it depends on
+  the choice of \(E_\alpha\) and \(F_\alpha\). Nevertheless, \(\mathfrak{h}\)
+  is stable under the action of \(\mathcal{W}\) -- i.e. \(\mathcal{W} \cdot
+  \mathfrak{h} \subset \mathfrak{h}\) -- and the restriction of this action to
+  \(\mathfrak{h}\) is independent of any choices.
+\end{note}
+
+See \cite[sec.~14.3]{humphreys} for a complete proof. Now the only thing we are
+missing for a complete classification is an existence and uniqueness theorem
+analogous to theorem~\ref{thm:sl2-exist-unique} and
 theorem~\ref{thm:sl3-existence-uniqueness}. It is already clear from the
 previous discussion that if \(\lambda\) is the highest weight of \(V\) then
 \(\lambda(H_\alpha) \ge 0\) for all positive roots \(\alpha\). Another way of