diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1,20 +1,19 @@
\chapter{Simple Weight Modules}\label{ch:mathieu}
-In this chapter we will expand our results on finite-dimensional simple
-modules of semisimple Lie algebras by generalizing them on multiple
-directions. First, we will now consider reductive Lie algebras, which means we
-can no longer take complete reducibility for granted. Namely, we have seen that
-if \(\mathfrak{g}\) is \emph{not} semisimple there must be some
+In this chapter we will expand our results on finite-dimensional simple modules
+of semisimple Lie algebras by generalizing them on multiple directions. First,
+we will now consider reductive Lie algebras, which means we can no longer take
+the semisimplicity of modules for granted. Namely, we have seen that if
+\(\mathfrak{g}\) is \emph{not} semisimple there must be some
\(\mathfrak{g}\)-module which is not the direct sum of simple
\(\mathfrak{g}\)-modules.
-Nevertheless, completely reducible \(\mathfrak{g}\)-modules are a \emph{very}
-large class of representations, and understanding them can still give us a lot
-of information regarding our algebra and the category of its modules --
-granted, not \emph{all} of the information as in the semisimple case. For this
-reason, we will focus exclusively on the classification of completely reducible
-modules. Our strategy is, once again, to classify the simple
-\(\mathfrak{g}\)-modules.
+Nevertheless, semisimple \(\mathfrak{g}\)-modules are a \emph{very} large class
+of representations, and understanding them can still give us a lot of
+information regarding our algebra and the category of its modules -- granted,
+not \emph{all} of the information as in the semisimple case. For this reason,
+we will focus exclusively on the classification of semisimple modules. Our
+strategy is, once again, to classify the simple \(\mathfrak{g}\)-modules.
Secondly, and this is more important, we now consider
\emph{infinite-dimensional} \(\mathfrak{g}\)-modules too, which introduces
@@ -171,12 +170,11 @@ isn't always the case. Nevertheless, in general we find\dots
\end{proposition}
This proof was deemed too technical to be included in here, but see Proposition
-3.5 of \cite{mathieu}. Again, there is plenty of examples of completely
-reducible modules which are \emph{not} weight modules. Nevertheless, weight
-modules constitute a large class of representations and understanding them can
-give us a lot of insight into the general case. Our goal is now classifying all
-simple weight \(\mathfrak{g}\)-modules for some fixed reductive Lie algebra
-\(\mathfrak{g}\).
+3.5 of \cite{mathieu}. Again, there is plenty of examples of semisimple modules
+which are \emph{not} weight modules. Nevertheless, weight modules constitute a
+large class of representations and understanding them can give us a lot of
+insight into the general case. Our goal is now classifying all simple weight
+\(\mathfrak{g}\)-modules for some fixed reductive Lie algebra \(\mathfrak{g}\).
As a first approximation of a solution to our problem, we consider the
induction functors \(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} :
@@ -535,13 +533,12 @@ follows.
for some \(\lambda \in \mathfrak{h}^*\).
\end{definition}
-Another natural candidate for the role of ``nice extensions'' are the completely
-reducible coherent families -- i.e. families which are completely reducible as
+Another natural candidate for the role of ``nice extensions'' are the
+semisimple coherent families -- i.e. families which are semisimple as
\(\mathfrak{g}\)-modules. These turn out to be very easy to produce. Namely,
-there is a construction, known as \emph{the semisimplification\footnote{Recall
-that a ``semisimple'' is a synonym for ``completely reducible'' in the context
-of modules.} of a coherent family}, which takes a coherent extension of \(M\)
-to a completely reducible coherent extension of \(M\).
+there is a construction, known as \emph{the semisimplification of a coherent
+family}, which takes a coherent extension of \(M\) to a semisimple coherent
+extension of \(M\).
% Mathieu's proof of this is somewhat profane, I don't think it's worth
% including it in here
@@ -552,9 +549,9 @@ to a completely reducible coherent extension of \(M\).
\begin{corollary}
Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a
- unique completely reducible coherent family
- \(\mathcal{M}^{\operatorname{ss}}\) of degree \(d\) such that the composition
- series of \(\mathcal{M}^{\operatorname{ss}}[\lambda]\) is the same as that of
+ unique semisimple coherent family \(\mathcal{M}^{\operatorname{ss}}\) of
+ degree \(d\) such that the composition series of
+ \(\mathcal{M}^{\operatorname{ss}}[\lambda]\) is the same as that of
\(\mathcal{M}[\lambda]\) for all \(\lambda \in \mathfrak{h}^*\), called
\emph{the semisimplification of \(\mathcal{M}\)}.
@@ -576,7 +573,7 @@ to a completely reducible coherent extension of \(M\).
\begin{proof}
The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear:
- since \(\mathcal{M}^{\operatorname{ss}}\) is completely reducible, so is
+ since \(\mathcal{M}^{\operatorname{ss}}\) is semisimple, so is
\(\mathcal{M}^{\operatorname{ss}}[\lambda]\). Hence by the Jordan-Hölder
Theorem
\[
@@ -591,7 +588,7 @@ to a completely reducible coherent extension of \(M\).
= \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}}
\mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}
\]
- is indeed a completely reducible coherent family of degree \(d\).
+ is indeed a semisimple coherent family of degree \(d\).
We know from examples~\ref{ex:submod-is-weight-mod} and
\ref{ex:quotient-is-weight-mod} that each quotient
@@ -678,12 +675,12 @@ promised, if \(\mathcal{M}\) is a coherent extension of \(M\) then so is
\end{proof}
Given the uniqueness of the semisimplification, the semisimplification of any
-completely reducible coherent extension \(\mathcal{M}\) is \(\mathcal{M}\)
+semisimple coherent extension \(\mathcal{M}\) is \(\mathcal{M}\)
itself and therefore\dots
\begin{corollary}\label{thm:admissible-is-submod-of-extension}
Let \(M\) be a simple admissible \(\mathfrak{g}\)-module and \(\mathcal{M}\)
- be a completely reducible coherent extension of \(M\). Then \(M\) is
+ be a semisimple coherent extension of \(M\). Then \(M\) is
contained in \(\mathcal{M}\).
\end{corollary}
@@ -692,9 +689,8 @@ well behaved coherent extensions. A complementary question now is: which
submodules of a \emph{nice} coherent family are cuspidal representations?
\begin{proposition}\label{thm:centralizer-multiplicity}
- Let \(M\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
- \(M_\lambda\) is a completely reducible
- \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
+ Let \(M\) be a semisimple weight \(\mathfrak{g}\)-module. Then \(M_\lambda\)
+ is a semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
\mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is the centralizer of
\(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\). Moreover, the multiplicity
of a given simple \(\mathfrak{g}\)-module \(L\) coincides with the
@@ -880,11 +876,11 @@ coherent extensions, which will be the focus of our next section.
\section{Localizations \& the Existence of Coherent Extensions}
-Let \(M\) be a simple admissible \(\mathfrak{g}\)-module of degree \(d\).
-Our goal is to prove that \(M\) has a (unique) irreducible completely reducible
-coherent extension \(\mathcal{M}\). Since \(M\) is simple, we know \(M
-\subset \mathcal{M}[\lambda]\) for any \(\lambda \in \operatorname{supp} M\).
-Our first task is constructing \(\mathcal{M}[\lambda]\). The issue here is that
+Let \(M\) be a simple admissible \(\mathfrak{g}\)-module of degree \(d\). Our
+goal is to prove that \(M\) has a (unique) irreducible semisimple coherent
+extension \(\mathcal{M}\). Since \(M\) is simple, we know \(M \subset
+\mathcal{M}[\lambda]\) for any \(\lambda \in \operatorname{supp} M\). Our first
+task is constructing \(\mathcal{M}[\lambda]\). The issue here is that
\(\operatorname{supp}_{\operatorname{ess}} M\) may not be all of \(\lambda + Q
= \operatorname{supp}_{\operatorname{ess}} \mathcal{M}[\lambda]\), so we may
find \(M \subsetneq \mathcal{M}[\lambda]\). In fact, we may find
@@ -1333,10 +1329,10 @@ It should now be obvious\dots
Lo and behold\dots
\begin{theorem}[Mathieu]
- There exists a unique completely reducible 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,
- \(\mExt(M)\) is a irreducible coherent family.
+ 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, \(\mExt(M)\)
+ is a irreducible coherent family.
\end{theorem}
\begin{proof}
@@ -1353,10 +1349,10 @@ Lo and behold\dots
M_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module for some
\(\lambda \in \operatorname{supp} M\).
- As for the uniqueness of \(\mExt(M)\), fix some other completely reducible
- coherent extension \(\mathcal{N}\) of \(M\). We claim that the multiplicity
- of a given simple \(\mathfrak{g}\)-module \(L\) in \(\mathcal{N}\) is
- determined by its \emph{trace function}
+ As for the uniqueness of \(\mExt(M)\), fix some other semisimple coherent
+ extension \(\mathcal{N}\) of \(M\). We claim that the multiplicity of a given
+ simple \(\mathfrak{g}\)-module \(L\) in \(\mathcal{N}\) is determined by its
+ \emph{trace function}
\begin{align*}
\mathfrak{h}^* \times \mathcal{U}(\mathfrak{g})_0 &
\to K \\
@@ -1365,8 +1361,8 @@ Lo and behold\dots
\end{align*}
It is a well known fact of the theory of modules that, given an associative
- \(K\)-algebra \(A\), a finite-dimensional completely reducible \(A\)-module
- \(L\) is determined, up to isomorphism, by its \emph{character}
+ \(K\)-algebra \(A\), a finite-dimensional semisimple \(A\)-module \(L\) is
+ determined, up to isomorphism, by its \emph{character}
\begin{align*}
\chi_L : A & \to K \\
a & \mapsto \operatorname{Tr}(a\!\restriction_L)
@@ -1403,14 +1399,14 @@ Lo and behold\dots
% \(\operatorname{Ext}(M)\) 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
-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.
+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, the problem of classifying \(\mathfrak{g}\)-family can be
reduced to that of classifying only \(\mathfrak{sl}_n(K)\)-families and
@@ -1446,7 +1442,7 @@ and \(\mathfrak{sp}_{2 n}(K)\) admit cuspidal modules, so it suffices to
consider these two cases.
Finally, we apply Mathieu's results to further reduce the problem to that of
-classifying the irreducible completely reducible coherent families of
+classifying 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