lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -113,8 +113,8 @@ to the case it holds. This brings us to the following definition.
   \left(\mfrac{V}{W}\right)_\lambda\) is surjective.
 \end{example}
 
-% TODOO: Prove this? Most likely not!
-% TODOO: Move this to somewhere else? Its kind of an akward place for this
+% TODOO: Move this to somewhere else? This probably fits the best just before
+% or after the first result that uses it
 \begin{proposition}\label{thm:centralizer-multiplicity}
   Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
   \(V_\lambda\) is a semisimple
@@ -190,43 +190,57 @@ Again, there is plenty of examples of completely reducible modules which are
 class of representations and understanding them can give us a lot of insight
 into the general case. Our goal is now classifying all irreducible weight
 \(\mathfrak{g}\)-modules for some fixed reductive Lie algebra \(\mathfrak{g}\).
-Historically, the first major step towards a solution to this classification
-problem wa given by Fernando in now infamous paper \citetitle{fernando}
-\cite{fernando}, in which he reduced our classification problem to that of
-classifying\dots
-
-\section{Cuspidal Representations}
-
-While remarkably engenious, Fernando's paper was based on the simple idea that
-we can reduce the classification problem by looking at induced representations.
-Namely, if \(\mathfrak{h}' \subset \mathfrak{g}\) is some proper subalgebra
-whose irreducible weight modules are known, we can produce lots of new weight
-\(\mathfrak{g}\)-modules by applying the induction function
-\(\operatorname{Ind}_{\mathfrak{h}'}^{\mathfrak{g}} :
-\mathfrak{h}'\text{-}\mathbf{Mod} \to \mathfrak{g}\text{-}\mathbf{Mod}\).
-Intuitively, if we recursively apply this ideas sufficiently many times the
-subalgebra \(\mathfrak{h}'\) will eventually become small enought for us
-directly classify its weight modules. This knowlage can then be used to extract
-information about the weight \(\mathfrak{g}\)-modules. We are particularly
-interested in the so called \emph{parabolic} subalgebras of \(\mathfrak{g}\),
-which we now define as follows.
+
+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
+\((\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V)_\lambda =
+\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{p})} V_\lambda\) for
+all \(\lambda \in \mathfrak{h}^*\). In particular,
+\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}\) takes weight
+\(\mathfrak{p}\)-modules to weight \(\mathfrak{g}\)-modules. This leads us to
+the following definition.
 
 \begin{definition}
   A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
   if \(\mathfrak{b} \subset \mathfrak{p}\).
 \end{definition}
 
-We should point out that while the representations induced by weight
-\(\mathfrak{p}\)-modules are weight \(\mathfrak{g}\)-modules, the module
-\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}(V)\) induced by some
-irreducible \(\mathfrak{p}\)-module \(V\) \emph{needs not} to be irreducible.
-Nevertheless, we can use it to produce an irreducible weight
-\(\mathfrak{g}\)-module via a construction very similar to that of Verma
-modules.
+% TODO: Define nilpotent algebras beforehand
+Parabolic subalgebras thus give us a process for constructing weight
+\(\mathfrak{g}\)-modules from representations of smaller (parabolic)
+subalgebras. Our hope is that by iterating this process again and again we can
+get a large class of irreducible weight \(\mathfrak{g}\)-modules. However,
+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 \(\mathfrak{u}\)
+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
+a reductive algebra.
+\begin{center}
+  \begin{tikzcd}
+    \mathfrak{p} \rar \dar & \mathfrak{gl}(V) \\
+    \mfrac{\mathfrak{p}}{\mathfrak{u}} \arrow[dotted]{ur} &
+  \end{tikzcd}
+\end{center}
+
+Let \(\mathfrak{p}\) be a parabolic subalgebra and \(V\) be an irreducible
+weight \(\mathfrak{p}\)-module. We should point out that while
+\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is a weight
+\(\mathfrak{g}\)-modules, it isn't necessarily irreducible. Nevertheless, we
+can use it to produce an irreducible weight \(\mathfrak{g}\)-module via a
+construction very similar to that of Verma modules.
 
 \begin{definition}
-  Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) and a
-  \(\mathfrak{p}\)-module \(V\) the module \(M_{\mathfrak{p}}(V) =
+  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.}.
@@ -251,29 +265,14 @@ This leads us to the following definitions.
   which is \emph{not} parabolic induced.
 \end{definition}
 
-% TODO: Define nilpotent algebras beforehand
-Let \(\mathfrak{p} \subset \mathfrak{g}\) be a parabolic subalgebra and
-\(\mathfrak{u} \subset \mathfrak{p}\) be the sum of all of its nilpotent ideals
--- i.e. a maximal nilpotent ideal, known as \emph{the nilradical of
-\(\mathfrak{p}\)}. The first observation we make is that the quotient
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\) is a reductive Lie algebra. Furthermore,
-one can show that if \(V\) is a weight \(\mathfrak{p}\)-module then
-\(\mathfrak{u}\) acts trivially in \(V\). By applying the universal property of
-quotients we can see that \(V\) has the natural structure of a
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\)-module.
-\begin{center}
-  \begin{tikzcd}
-    \mathfrak{p} \rar \dar & \mathfrak{gl}(V) \\
-    \mfrac{\mathfrak{p}}{\mathfrak{u}} \arrow[dotted]{ur} &
-  \end{tikzcd}
-\end{center}
-
-In particular, it makes sence to call a weight \(\mathfrak{p}\)-module
+Since every weight \(\mathfrak{p}\)-module \(V\) is an
+\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\)-module, it makes sence to call \(V\)
 \emph{cuspidal} if it is a cuspidal representation of
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\). Fernando's great breaktrought was the
-realization that \emph{all} irreducible weight \(\mathfrak{g}\)-modules are
-parabolic induced modules induced by cuspidal \(\mathfrak{p}\)-modules. In
-other words\dots
+\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\). Historically, the first breaktrought
+regarding our classification problem was given by Fernando in his now infamous
+paper \citetitle{fernando} \cite{fernando}, where he proved that every
+irreducible weight \(\mathfrak{g}\)-module is parabolic induced. In other
+words\dots
 
 \begin{theorem}[Fernando]
   Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to
@@ -281,8 +280,6 @@ other words\dots
   \mathfrak{g}\) and some irreducible cuspidal \(\mathfrak{p}\)-module \(V\).
 \end{theorem}
 
-The significance of this result should be self-evident: as promised, we've now
-reduced the classification problem to classifying the cuspidal representations.
 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