lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -1,2 +1,54 @@
 \chapter{Classification of Coherent Families}
 
+% This is a very important theorem, but since we won't classify the coherent
+% extensions in here we don't need it, and there is no other motivation behind
+% it. Including this would also require me to explain what central characters
+% are, which is a bit of a pain
+%\begin{proposition}[Mathieu]
+%  The central characters of the irreducible submodules of
+%  \(\operatorname{Ext}(M)\) are all the same.
+%\end{proposition}
+
+First and foremost, notice that because of
+Example~\ref{thm:simple-weight-mod-is-tensor-prod} the problem of classifying
+the simple weight \(\mathfrak{g}\)-modules can be reduced to that of
+classifying the simple weight modules of its simple components. In addition, it
+turns out that very few simple Lie algebras admit cuspidal modules at all.
+Specifically\dots
+
+\begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
+  Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra. Suppose
+  there exists a cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s} \cong
+  \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\).
+\end{proposition}
+
+% TODO: Remove this: we will only focus on the combinatorial classification
+% TODO: Simply notice that a more explicit "geometric" description of the
+% cohorent families exists
+Hence it suffices to classify the irreducible semisimple coherent families of
+\(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\). These can be described
+either algebraically, using combinatorial invariants -- which Mathieu does in
+sections 7, 8 and 9 of his paper -- or geometrically, using algebraic varieties
+and differential forms -- which is done in sections 11 and 12. While rather
+complicated on its own, the geometric construction is more concrete than its
+combinatorial counterpart.
+
+% TODO: Change this
+% I don't really think these notes bring us to this conclusion
+% If anything, these notes really illustrate the incredible vastness of the
+% ocean of representation theory, how unknowable it is, and the remarkable
+% amount of engenuity required to explore it
+This construction also brings us full circle to the beginning of these notes,
+where we saw in Proposition~\ref{thm:geometric-realization-of-uni-env} that
+\(\mathfrak{g}\)-modules may be understood as geometric objects. In fact,
+throughout the previous four chapters we have seen a tremendous number of
+geometrically motivated examples, which further emphasizes the connection
+between representation theory and geometry. I would personally go as far as
+saying that the beautiful interplay between the algebraic and the geometric is
+precisely what makes representation theory such a fascinating and charming
+subject.
+
+Alas, our journey has come to an end. All it is left is to wonder at the beauty
+of Lie algebras and their representations.
+
+\label{end-47}

diff --git a/sections/simple-weight.tex b/sections/simple-weight.tex
@@ -926,7 +926,7 @@ family are cuspidal?
 
 To finish the proof, we now show\dots
 
-\begin{lemma}
+\begin{lemma}\label{thm:set-of-simple-u0-mods-is-open}
   Let \(\mathcal{M}\) be a coherent family. The set \(U = \{\lambda \in
   \mathfrak{h}^* : \mathcal{M}_\lambda \ \text{is a simple
   $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is Zariski-open.
@@ -1506,7 +1506,7 @@ It should now be obvious\dots
 
 Lo and behold\dots
 
-\begin{theorem}[Mathieu]\index{coherent family!Mathieu's \(\mExt\) coherent extension}
+\begin{theorem}[Mathieu]\label{thm:mathieu-ext-exists-unique}\index{coherent family!Mathieu's \(\mExt\) coherent extension}
   There exists a unique semisimple coherent extension \(\mExt(M)\) of \(M\).
   More precisely, if \(\mathcal{M}\) is any coherent extension of \(M\), then
   \(\mathcal{M}^{\operatorname{ss}} \cong \mExt(M)\). Furthermore, \(\mExt(M)\)
@@ -1568,56 +1568,36 @@ Lo and behold\dots
   In conclusion, \(\mathcal{N} \cong \mExt(M)\) and \(\mExt(M)\) is unique.
 \end{proof}
 
-% This is a very important theorem, but since we won't classify the coherent
-% extensions in here we don't need it, and there is no other motivation behind
-% it. Including this would also require me to explain what central characters
-% are, which is a bit of a pain
-%\begin{proposition}[Mathieu]
-%  The central characters of the irreducible submodules of
-%  \(\operatorname{Ext}(M)\) are all the same.
-%\end{proposition}
-
-We have thus concluded our classification of cuspidal modules in terms of
-coherent families. Of course, to get an explicit construction of all simple
-\(\mathfrak{g}\)-modules we would have to classify the irreducible semisimple
-coherent \(\mathfrak{g}\)-families themselves, which is the subject of sections
-7, 8 and 9 of \cite{mathieu}. In addition, in sections 11 and 12 of
-\cite{mathieu} Mathieu provides an explicit construction of coherent families.
-We unfortunately do not have the necessary space to discuss these results in
-detail, but we will now provide a brief overview.
-
-First and foremost, notice that because of
-Example~\ref{thm:simple-weight-mod-is-tensor-prod} the problem of classifying
-the simple weight \(\mathfrak{g}\)-modules can be reduced to that of
-classifying the simple weight modules of its simple components. In addition, it
-turns out that very few simple Lie algebras admit cuspidal modules at all.
-Specifically\dots
-
-\begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
-  Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra. Suppose
-  there exists a cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s} \cong
-  \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\).
+A sort of ``reciprocal'' of Theorem~\ref{thm:mathieu-ext-exists-unique} also
+holds. Namely\dots
+
+\begin{proposition}
+  Let \(\mathcal{M}\) be a semisimple irreducible coherent extension. Then
+  there exists some simple bounded \(\mathfrak{g}\)-module \(M\) such that
+  \(\mathcal{M} \cong \mExt(M)\).
 \end{proposition}
 
-Hence it suffices to classify the irreducible semisimple coherent families of
-\(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\). These can be described
-either algebraically, using combinatorial invariants -- which Mathieu does in
-sections 7, 8 and 9 of his paper -- or geometrically, using algebraic varieties
-and differential forms -- which is done in sections 11 and 12. While rather
-complicated on its own, the geometric construction is more concrete than its
-combinatorial counterpart.
-
-This construction also brings us full circle to the beginning of these notes,
-where we saw in Proposition~\ref{thm:geometric-realization-of-uni-env} that
-\(\mathfrak{g}\)-modules may be understood as geometric objects. In fact,
-throughout the previous four chapters we have seen a tremendous number of
-geometrically motivated examples, which further emphasizes the connection
-between representation theory and geometry. I would personally go as far as
-saying that the beautiful interplay between the algebraic and the geometric is
-precisely what makes representation theory such a fascinating and charming
-subject.
-
-Alas, our journey has come to an end. All it is left is to wonder at the beauty
-of Lie algebras and their representations.
-
-\label{end-47}
+\begin{proof}
+  Let \(M \subset \mathcal{M}\) be any simple submodule and \(d = \deg
+  \mathcal{M}\). Since \(M\) is bounded,
+  \(\operatorname{supp}_{\operatorname{ess}} M\) is Zariski-dense. In addition,
+  it follows from Lemma~\ref{thm:set-of-simple-u0-mods-is-open} that \(U =
+  \{\lambda \in \mathfrak{h}^* : \mathcal{M}_\lambda \ \text{is a simple
+  $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is a Zariski-open subset
+  -- which is non-empty since \(\mathcal{M}\) is irreducible.
+
+  Hence there is some \(\lambda \in \operatorname{supp}_{\operatorname{ess}} M
+  \cap U\). In particular, there is some \(\lambda \in
+  \operatorname{supp}_{\operatorname{ess}} M\) such that \(M_\lambda =
+  \mathcal{M}_\lambda\) and thus \(\deg M = \dim \mathcal{M}_\lambda = d\).
+  This implies that \(\mathcal{M}\) is a coherent extension of \(M\), so that
+  by the uniqueness of semisimple irreducible coherent extensions we get
+  \(\mathcal{M} \cong \mExt(M)\).
+\end{proof}
+
+Having thus reduced the problem of classifying the cuspidal
+\(\mathfrak{g}\)-modules to that of understanding semisimple irreducible
+coherent families, the only remaining question for us to tackle is: what are
+the coherent \(\mathfrak{g}\)-families? This turns out to be a decently
+complicated question on its own, and we will require a full chapter to answer
+it. This will be the focus of our final chapter.