lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -194,11 +194,11 @@ into the general case. Our goal is now classifying all irreducible weight
 As a first approximation of a solution to our problem, we consider the
 induction functors \(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} :
 \mathfrak{p}\text{-}\mathbf{Mod} \to \mathfrak{g}\text{-}\mathbf{Mod}\), where
-\(\mathcal{p} \subset \mathfrak{g}\) is some subalgebra. These functors have
-already proved themselves a powerful tool for constructing representations. Our
-first observation is that if \(\mathfrak{p} \subset \mathfrak{g}\) contains the
-Borel \(\mathfrak{b}\) then \(\mathfrak{h} \subset \mathfrak{p}\) is a Cartan
-subalgebra of \(\mathfrak{p}\) and
+\(\mathfrak{p} \subset \mathfrak{g}\) is some subalgebra. These functors have
+already proved themselves a powerful tool for constructing representations in
+the previous chapters. Our first observation is that if \(\mathfrak{p} \subset
+\mathfrak{g}\) contains the Borel \(\mathfrak{b}\) then \(\mathfrak{h} \subset
+\mathfrak{p}\) is a Cartan subalgebra of \(\mathfrak{p}\) and
 \((\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V)_\lambda =
 \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{p})} V_\lambda\) for
 all \(\lambda \in \mathfrak{h}^*\). In particular,
@@ -220,7 +220,7 @@ there's a small catch: a parabolic subalgebra \(\mathfrak{p} \subset
 \mathfrak{g}\) needs not to be reductive. We can get around this limitation by
 considering the sum \(\mathfrak{u} \normal \mathfrak{p}\) of all of its
 nipotent ideals -- i.e. a maximal nilpotent ideal of \(\mathfrak{p}\), known as
-\emph{the nilradical of \(\mathfrak{p}\)} and noticing that \(\mathfrak{u}\)
+\emph{the nilradical of \(\mathfrak{p}\)} -- and noticing that \(\mathfrak{u}\)
 acts trivialy in any weight \(\mathfrak{p}\)-module \(V\). By applying the
 universal property of quotients we can see that \(V\) has the natural structure
 of a representation of \(\mfrac{\mathfrak{p}}{\mathfrak{u}}\), which is always
@@ -283,8 +283,8 @@ words\dots
 We should point out that the relationship between irreducible weight
 \(\mathfrak{g}\)-modules and pairs \((\mathfrak{p}, V)\) -- where
 \(\mathfrak{p}\) is some parabolic subalgebra and \(V\) is an irreducible
-cuspidal \(\mathfrak{p}\)-module -- is not one-to-one. In fact, one can
-show\dots
+cuspidal \(\mathfrak{p}\)-module -- is not one-to-one. Nevertheless, this
+relationship is well understood. Namely, Fernando himself established\dots
 
 % TODO: Define the conjugation of a p-mod by an element of the Weil group
 % beforehand