lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -283,12 +283,13 @@ relationship is well understood. Namely, Fernando himself established\dots
   \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
+  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
   \[
     \mathcal{W}_V
     = \langle
-      T_\beta : \beta \in \Sigma, H_\beta + \mathfrak{u}
+      \sigma_\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

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -519,30 +519,31 @@ in the \(\mathfrak{sl}_2(K)\)-module \(\bigoplus_k V_{\lambda - k \alpha}\).
 
 This has a number of important consequences. For instance\dots
 
-% TODO: Change the notation for T_alpha
 \begin{corollary}
-  If \(\alpha \in \Delta^+\) and \(T_\alpha : \mathfrak{h}^* \to
+  If \(\alpha \in \Delta^+\) and \(\sigma_\alpha : \mathfrak{h}^* \to
   \mathfrak{h}^*\) is the reflection in the hyperplane perpendicular to
   \(\alpha\) with respect to the Killing form, the weights of \(V\) occuring in
-  the line joining \(\lambda\) and \(T_\alpha\) are precisely the \(\mu \in P\)
-  lying between \(\lambda\) and \(T_\alpha \lambda\).
+  the line joining \(\lambda\) and \(\sigma_\alpha\) are precisely the \(\mu
+  \in P\) lying between \(\lambda\) and \(\sigma_\alpha(\lambda)\).
 \end{corollary}
 
 \begin{proof}
-  Notice that any \(\mu \in P\) in the line joining \(\lambda\) and \(T_\alpha
-  \lambda\) has the form \(\mu = \lambda - k \alpha\) for some \(k\), so that
-  \(V_\mu\) corresponds the eigenspace associated with the eigenvalue
-  \(\lambda(H_\alpha) - 2k\) of the action of \(h\) in \(\bigoplus_k V_{\lambda
-  - k \alpha}\). If \(\mu\) lies between \(\lambda\) and \(T_\alpha \lambda\)
-  then \(k\) lies between \(0\) and \(\lambda(H_\alpha)\), in which case
-  \(V_\mu \neq 0\) and therefore \(\mu\) is a weight.
-
-  On the other hand, if \(\mu\) does not lie between \(\lambda\) and \(T_\alpha
-  \lambda\) then either \(k < 0\) or \(k > \lambda(H_\alpha)\). Suppose \(\mu\)
-  is a weight. In the first case \(\mu \succ \lambda\), a contradiction. On the
-  second case the fact that \(V_\mu \ne 0\) implies \(V_{\lambda  + (k -
-  \lambda(H_\alpha)) \alpha} \ne 0 = V_{T_\alpha \mu}\), which contradicts the
-  fact that \(V_{\lambda + \ell \alpha} = 0\) for all \(\ell \ge 0\).
+  Notice that any \(\mu \in P\) in the line joining \(\lambda\) and
+  \(\sigma_\alpha(\lambda)\) has the form \(\mu = \lambda - k \alpha\) for some
+  \(k\), so that \(V_\mu\) corresponds the eigenspace associated with the
+  eigenvalue \(\lambda(H_\alpha) - 2k\) of the action of \(h\) in \(\bigoplus_k
+  V_{\lambda - k \alpha}\). If \(\mu\) lies between \(\lambda\) and
+  \(\sigma_\alpha(\lambda)\) then \(k\) lies between \(0\) and
+  \(\lambda(H_\alpha)\), in which case \(V_\mu \neq 0\) and therefore \(\mu\)
+  is a weight.
+
+  On the other hand, if \(\mu\) does not lie between \(\lambda\) and
+  \(\sigma_\alpha(\lambda)\) then either \(k < 0\) or \(k >
+  \lambda(H_\alpha)\). Suppose \(\mu\) is a weight. In the first case \(\mu
+  \succ \lambda\), a contradiction. On the second case the fact that \(V_\mu
+  \ne 0\) implies \(V_{\lambda  + (k - \lambda(H_\alpha)) \alpha} =
+  V_{\sigma_\alpha(\mu)} \ne 0\), which contradicts the fact that \(V_{\lambda
+  + \ell \alpha} = 0\) for all \(\ell \ge 0\).
 \end{proof}
 
 This is entirely analogous to the situation of \(\mathfrak{sl}_3(K)\), where we
@@ -552,9 +553,9 @@ string symmetric with respect to the lines \(K \alpha\) with \(B(\alpha_i -
 class of arguments leads us to the conclusion\dots
 
 \begin{definition}
-  We refer to the group \(\mathcal{W} = \langle T_\alpha : \alpha \in \Delta^+
-  \rangle \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl group of
-  \(\mathfrak{g}\)}.
+  We refer to the group \(\mathcal{W} = \langle \sigma_\alpha : \alpha \in
+  \Delta^+ \rangle \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl
+  group of \(\mathfrak{g}\)}.
 \end{definition}
 
 \begin{theorem}\label{thm:irr-weight-class}