diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -216,6 +216,7 @@ the following definition.
\end{definition}
% TODO: Define nilpotent algebras beforehand
+% TODO: State the universal property of quotients in the introduction
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
@@ -272,11 +273,10 @@ This leads us to the following definitions.
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
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\). 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
+\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\). 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
@@ -356,14 +356,136 @@ characterizations of cuspidal modules.
% is an irreducible weight \(\mathfrak{s}_i\)-module.
%\end{proposition}
+Having reduced our classification problem to that o classifying irreducible
+cuspidal representations, we are now faced the dauting task of actually
+classifying them. Historically, this was first achieved by Olivier Mathieu in
+the early 2000's in his paper \citetitle{mathieu} \cite{mathieu}. To do so,
+Mathieu introduced new tools which have since proved themselves remarkably
+usefull troughtout the field, known as\dots
+
\section{Coherent Families}
+We begin our analysis with a simple question: how to do we go about contructing
+cuspidal representations? Specifically, given a cuspidal
+\(\mathfrak{g}\)-module, how can we use it to produce new cuspidal
+representations? To answer this question, we look back at the single example of
+a cuspidal representations we have encoutered so far: the
+\(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\) of Laurent polynomials -- i.e.
+example~\ref{ex:laurent-polynomial-mod}.
+
+% TODO: Add a definition of the ring of differential operators somewhere?
+Our first observation is that \(\mathfrak{sl}_2(K)\) acts in \(K[x, x^{-1}]\)
+via differential operators. In other words, the action map
+\(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{End}(K[x, x^{-1}])\)
+factors trought the inclusion of the algebra \(\operatorname{Diff}(K[x,
+x^{-1}]) = K\left[x, x^{-1}, \frac{\mathrm{d}}{\mathrm{d}x}\right]\) of
+differential operators in \(K[x, x^{-1}]\).
+\begin{center}
+ \begin{tikzcd}
+ \mathcal{U}(\mathfrak{sl}_2(K)) \rar &
+ \operatorname{Diff}(K[x, x^{-1}]) \rar &
+ \operatorname{End}(K[x, x^{-1}])
+ \end{tikzcd}
+\end{center}
+
+The space \(K[x, x^{-1}]\) can be regarded as a \(\operatorname{Diff}(K[x,
+x^{-1}])\)-module in the natural way, and we can produce new
+\(\operatorname{Diff}(K[x, x^{-1}])\)-modules by twisting \(K[x, x^{-1}]\) by
+automorphisms of \(\operatorname{Diff}(K[x, x^{-1}])\). For example, given
+\(\lambda \in K\) we may take the automorphism
+\begin{align*}
+ \varphi_\lambda : \operatorname{Diff}(K[x, x^{-1}]) &
+ \to \operatorname{Diff}(K[x, x^{-1}]) \\
+ x & \mapsto x \\
+ x^{-1} & \mapsto x^{-1} \\
+ \frac{\mathrm{d}}{\mathrm{d} x} & \mapsto \frac{\mathrm{d}}{\mathrm{d} x} +
+ \frac{\lambda}{2} x^{-1}
+\end{align*}
+and consider the module \(\varphi_\lambda K[x, x^{-1}] = K[x, x^{-1}]\) where
+some operator \(L \in \operatorname{Diff}(K[x, x^{-1}])\) acts as
+\(\varphi_\lambda(L)\).
+
+By composing the action map \(\operatorname{Diff}(K[x, x^{-1}]) \to
+\operatorname{End}(\varphi_\lambda K[x, x^{-1}])\) with the homomorphism of
+algebras \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x,
+x^{-1}])\) we can give \(\varphi_\lambda K[x, x^{-1}]\) the structure of an
+\(\mathfrak{sl}_2(K)\)-module. Diagramaticaly, we have
+\begin{center}
+ \begin{tikzcd}
+ \mathcal{U}(\mathfrak{sl}_2(K)) \rar &
+ \operatorname{Diff}(K[x, x^{-1}]) \rar{\varphi_\lambda} &
+ \operatorname{Diff}(K[x, x^{-1}]) \rar &
+ \operatorname{End}(K[x, x^{-1}])
+ \end{tikzcd},
+\end{center}
+where the maps \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x,
+x^{-1}])\) and \(\operatorname{Diff}(K[x, x^{1}]) \to \operatorname{End}(K[x,
+x^{-1}])\) are the ones from the previous diagram.
+
+% TODO: Point out that the twisting automorphism does not fact to an
+% automorphism of the universal enveloping algebra of sl2, but it factor
+% trought an automorphism of the localization of this algebra by f
+
+Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) in
+\(\varphi_\lambda K[x, x^{-1}]\) is given by
+\begin{align*}
+ p & \overset{f}{\mapsto}
+ \left(
+ - \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1 - \lambda}{2} x^{-1}
+ \right) p &
+ p & \overset{h}{\mapsto}
+ \left( 2 x \frac{\mathrm{d}}{\mathrm{d}x} + \lambda \right) p &
+ p & \overset{e}{\mapsto}
+ \left(
+ x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1 + \lambda}{2} x
+ \right) p,
+\end{align*}
+so we can see \((\varphi_\lambda K[x, x^{-1}])_{2 k + \frac{\lambda}{2}} = K
+x^k\) for all \(k \in \mathbb{Z}\) and \((\varphi_\lambda K[x, x^{-1}])_\mu =
+0\) for all other \(\mu \in \mathfrak{h}^*\).
+
+Hence \(\varphi_\lambda K[x, x^{-1}]\)
+is a degree \(1\) admissible \(\mathfrak{sl}_2(K)\)-module with
+\(\operatorname{supp} \varphi_\lambda K[x, x^{-1}] = \frac{\lambda}{2} + 2
+\mathbb{Z}\). One can also quickly check that if \(\lambda \notin 1 + 2
+\mathbb{Z}\) then \(e\) and \(f\) act injectively in \(\varphi_\lambda K[x,
+x^{-1}]\), so that \(\varphi_\lambda K[x, x^{-1}]\) is irreducible. In
+particular, if \(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with \(\lambda \notin
+\mu + 2 \mathbb{Z}\) then \(\varphi_\lambda K[x, x^{-1}]\) and \(\varphi_\mu
+K[x, x^{-1}]\) are non-isomorphic irreducible cuspidal \(\mathfrak{sl}_2(K)\),
+since their supports differ.
+These cuspidal representations can be ``glued toghether'' in a \emph{monstrous
+concoction} by summing over \(\lambda \in K\), as in
+\[
+ \mathcal{M}
+ = \bigoplus_{\lambda + 2 \mathbb{Z} \in \mfrac{K}{2 \mathbb{Z}}}
+ \varphi_\lambda K[x, x^{-1}],
+\]
+
+To a distracted spectator, \(\mathcal{M}\) may look like just another, inocent,
+\(\mathfrak{sl}_2(K)\)-module. However, the attentitive reader may have already
+noticed some of the its bizare features, most noticeable of which is the fact
+that \(\mathcal{M}\) is very big. In fact, \(\mathcal{M}\) is as big a degree
+\(1\) admissible representation gets: \(\operatorname{supp} \mathcal{M} =
+\operatorname{supp}_{\operatorname{ess}} \mathcal{M}\) is the entirety of
+\(\mathfrak{h}^*\). This should look very alian to readers familiarized with
+the theory of finite-dimensional weight modules. For this reason alone,
+\(\mathcal{M}\) deserves to be called ``a monstruous concoction''.
+
+On a perhaps less derogatory note, \(\mathcal{M}\) also deserves to be called
+\emph{a family}. This is because \(\mathcal{M}\) consists of lots of smaller
+cuspidal representations which fit together inside of it in a \emph{coherent}
+fashion. Mathieu's engineous breaktrough was the realization that
+\(\mathcal{M}\) is a particular example of a more general pattern, which ha
+named \emph{coherent families}.
+
\begin{definition}
A \emph{coherent family \(\mathcal{M}\) of degree \(d\)} is a weight
\(\mathfrak{g}\)-module \(\mathcal{M}\) such that
\begin{enumerate}
\item \(\dim \mathcal{M}_\lambda = d\) for \emph{all} \(\lambda \in
- \mathfrak{h}^*\)
+ \mathfrak{h}^*\) -- i.e. \(\operatorname{supp}_{\operatorname{ess}}
+ \mathcal{M} = \mathfrak{h}^*\)
\item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the centralizer
\(\mathcal{U}(\mathfrak{g})_0\) of \(\mathfrak{h}\) in
\(\mathcal{U}(\mathfrak{g})\), the map
@@ -376,11 +498,25 @@ characterizations of cuspidal modules.
\end{enumerate}
\end{definition}
-% TODO: Add an example: there's an example of a coherent sl2-family in
-% Mathieu's paper
-% TODO: Add a discussion on how this may sound unintuitive, but the motivation
-% comes from the relationship between highest weight modules and coherent
-% families
+\begin{example}\label{ex:sl-laurent-family}
+ The module \(\mathcal{M} = \bigoplus_{\lambda + 2 \mathbb{Z} \in
+ \mfrac{K}{2 \mathbb{Z}}} \varphi_\lambda K[x, x^{-1}]\) is a degree \(1\)
+ coherent \(\mathfrak{sl}_2(K)\)-family.
+\end{example}
+
+\begin{example}
+ Given \(\lambda \in K\), \(\mathcal{M}(\lambda) = \bigoplus_{\mu \in K} K
+ m_\mu\) with
+ \begin{align*}
+ m_\mu & \overset{f}{\mapsto} (\lambda - \mu) m_{\mu - 1} &
+ m_\mu & \overset{h}{\mapsto} 2\mu m_\mu &
+ m_\mu & \overset{e}{\mapsto} (\lambda + \mu) m_{\mu + 1},
+ \end{align*}
+ is a degree \(1\) coherent \(\mathfrak{sl}_2(K)\)-family. It is easy to check
+ \(\mathcal{M}\) from example~\ref{ex:sl-laurent-family} is isomorphic to
+ \(\mathcal{M}(\sfrac{1}{2})\). In fact by rewriting the symbol \(m_k\) as
+ \(x^k\) we can see \((\mathcal{M}(\sfrac{1}{2}))[0] \cong K[x, x^{-1}]\).
+\end{example}
% TODO: Point out this is equivalent to M being a simple object in the
% category of coherent families