diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -190,19 +190,46 @@ 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.
\begin{definition}
A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
if \(\mathfrak{b} \subset \mathfrak{p}\).
\end{definition}
-% TODO: Comment afterwords that the Verma modules are indeed generalized Verma
-% modules
+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.
+
\begin{definition}
Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) and a
\(\mathfrak{p}\)-module \(V\) the module \(M_{\mathfrak{p}}(V) =
\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is called \emph{a
- generalized Verma module}.
+ generalized Verma module}\footnote{It should be clear from the definitions
+ that Verma modules are indeed generalized Verma modules.}.
\end{definition}
\begin{proposition}
@@ -213,6 +240,8 @@ into the general case. Our goal is now classifying all irreducible weight
The irreducible quotient \(L_{\mathfrak{p}}(V)\) is a weight module.
\end{proposition}
+This leads us to the following definitions.
+
\begin{definition}
An irreducible \(\mathfrak{g}\)-module is called \emph{parabolic induced} if
it is isomorphic to \(L_{\mathfrak{p}}(V)\) for some proper parabolic
@@ -222,23 +251,51 @@ into the general case. Our goal is now classifying all irreducible weight
which is \emph{not} parabolic induced.
\end{definition}
-% TODO: Remark on the fact that any simple weight p-mod is a (p/u)-mod, so that
-% the notation of a cuspidal p-mod is well definited
-% TODO: Define the conjugation of a p-mod by an element of the Weil group?
+% 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
+\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
+
\begin{theorem}[Fernando]
Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to
\(L_{\mathfrak{p}}(V)\) for some parabolic subalgebra \(\mathfrak{p} \subset
\mathfrak{g}\) and some irreducible cuspidal \(\mathfrak{p}\)-module \(V\).
\end{theorem}
-% TODO: Point out that the relationship between p-modules and cuspidal
-% g-modules is not 1-to-1
+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
+cuspidal \(\mathfrak{p}\)-module -- is not one-to-one. In fact, one can
+show\dots
+% TODO: 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 \(\Sigma \subset
- \Delta_{\mathfrak{p}_1}\). Furthermore, if \(\mathfrak{p}' \subset
+ 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
@@ -255,11 +312,14 @@ into the general case. Our goal is now classifying all irreducible weight
\]
\end{proposition}
-% TODO: Point out that the definition of W_V is independant of the choice of
-% Sigma
+\begin{note}
+ The definition of the subgroup \(\mathcal{W}_V \subset \mathcal{W}\) is
+ independant of the choice of basis \(\Sigma\).
+\end{note}
+
+As a first consequence of Fernando's theorem, we provide two alternative
+characterizations of cuspidal modules.
-% TODO: Remark that the support of a simple weight module is always contained
-% in a coset
\begin{corollary}[Fernando]\label{thm:cuspidal-mod-equivs}
Let \(V\) be an irreducible weight \(\mathfrak{g}\)-module. The following
conditions are equivalent.
@@ -281,19 +341,19 @@ into the general case. Our goal is now classifying all irreducible weight
\(\mathfrak{sl}_2(K)\).
\end{example}
-% TODOO: Do we need this proposition? I think this only comes up in the
-% classification of simple completely reducible coherent families
-\begin{proposition}
- If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
- \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
- and \(\mathfrak{s}_i\) is a simple component of \(\mathfrak{g}\), then any
- irreducible weight \(\mathfrak{g}\)-module \(V\) decomposes as
- \[
- V = Z \otimes V_1 \otimes \cdots \otimes V_n
- \]
- where \(Z\) is a 1-dimensional representation of \(\mathfrak{z}\) and \(V_i\)
- is an irreducible weight \(\mathfrak{s}_i\)-module.
-\end{proposition}
+%% TODOO: Do we need this proposition? I think this only comes up in the
+%% classification of simple completely reducible coherent families
+%\begin{proposition}
+% If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
+% \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
+% and \(\mathfrak{s}_i\) is a simple component of \(\mathfrak{g}\), then any
+% irreducible weight \(\mathfrak{g}\)-module \(V\) decomposes as
+% \[
+% V = Z \otimes V_1 \otimes \cdots \otimes V_n
+% \]
+% where \(Z\) is a 1-dimensional representation of \(\mathfrak{z}\) and \(V_i\)
+% is an irreducible weight \(\mathfrak{s}_i\)-module.
+%\end{proposition}
\section{Coherent Families}