lie-algebras-and-their-representations

diff --git a/references.bib b/references.bib
@@ -215,3 +215,12 @@
   doi =     {https://doi.org/10.1016/j.jpaa.2015.09.013},
   author =  {Jonathan Nilsson},
 }
+
+@article{fernando,
+  title =   {Lie algebra modules with finite-dimensional weight spaces. I},
+  author =  {S Rupasiri Lakshman Fernando},
+  journal = {Transactions of the American Mathematical Society},
+  year =    {1990},
+  volume =  {322},
+  pages =   {757-781},
+}

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}