lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -327,20 +327,6 @@ characterizations of cuspidal modules.
   \(\mathfrak{sl}_2(K)\).
 \end{example}
 
-%% TODO: Move this to the section where we discuss the classification of
-%% 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}
-
 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
@@ -1404,4 +1390,65 @@ Lo and behold\dots
 %  \(\operatorname{Ext}(V)\) are all the same.
 %\end{proposition}
 
-% TODOO: Write a conclusion
+We have thus concluded our classification of cuspidal representations in terms
+of coherent families. Of course, to get an explicit construction of all
+irreducible \(\mathfrak{g}\)-modules we would have to classify the simple
+completely reducible 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 formost, the problem of classifying the coherent
+\(\mathfrak{g}\)-family can be reduced to that of classifying only the coherent
+\(\mathfrak{sl}_n(K)\)-families and coherent \(\mathfrak{sp}_{2
+n}(K)\)-families. This is because of the following results.
+
+% TODO: Define V boxtimes W
+\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 \(V\) can be decomposed as
+  \[
+    V \cong Z \boxtimes V_1 \boxtimes \cdots \boxtimes V_n
+  \]
+  where \(Z\) is a 1-dimensional representation of \(\mathfrak{z}\) and \(V_i\)
+  is an irreducible weight \(\mathfrak{s}_i\)-module.
+\end{proposition}
+
+\begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
+  Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra. Suppose
+  there exists an irreducible cuspidal \(\mathfrak{s}\)-module. Then
+  \(\mathfrak{s} \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong
+  \mathfrak{sp}_{2 n}(K)\).
+\end{proposition}
+
+We've previously seen that the representation of Abelian Lie algebras are well
+understood, so to classify the irreducible representations of an arbitrary
+reductive algebra it suffices to classify those of its simple components. To
+classify these representations we can apply Fernando's results and reduce the
+problem to constructing the cuspidal representation of the simple Lie algebras.
+But by proposition~\ref{thm:only-sl-n-sp-have-cuspidal} only
+\(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\) admit cuspidal
+representation, so it suffices to consider these two cases.
+
+Finally, we apply Mathieu's results to further reduce the problem to that of
+classifying the simple completely reducible coherent families of
+\(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\). These can be described
+either algebraicaly, using combinatorial invariants -- which Mathieu does in
+sections 7, 8 and 9 of his paper -- or geometricaly, using affine algebraic
+varieties -- 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 beggining 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 our journey 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 beutiful interplay between the algebraic and the geometric is precisely
+what makes representation theory such a charming subject.
+
+% TODO: Falar uma frase de efeito!