- Commit
- 0d471bdccd9c35c746b87702bcea033aa60b56e6
- Parent
- 2b0f999844861c7b3a2570ce8a25614998f04205
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Corrected the last chapter
Corrected the proofs and statements of the last chapter to account for the case where 𝔤 is semisimple but not simple
This case is dealt-with by Mathieu
Added many examples in the proccess
Also added many TODO items and corrected a few typos
diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1,29 +1,21 @@
\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
-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, 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
-numerous complications to our analysis. For example, if
-\(\mathcal{U}(\mathfrak{g})\) is the regular \(\mathfrak{g}\)-module then
-\(\mathcal{U}(\mathfrak{g})_\lambda = 0\) for all \(\lambda \in
-\mathfrak{h}^*\). This follows from the fact that \(\mathcal{U}(\mathfrak{g})\)
-has no zero divisors: given \(u \in \mathcal{U}(\mathfrak{g})\), \((H -
-\lambda(H)) u = 0\) for some nonzero \(H \in \mathfrak{h}\) implies \(u = 0\).
-In particular,
+of semisimple Lie algebras by considering \emph{infinite-dimensional}
+\(\mathfrak{g}\)-modules, which introduces numerous complications to our
+analysis.
+
+For instance, in the infinite-dimensional setting we can no longer take
+complete-reducibility for granted. Indeed, we have seen that even if
+\(\mathfrak{g}\) is a semisimple Lie algebra, there are infinite-dimensional
+\(\mathfrak{g}\)-modules which are not semisimple. For a counterexample look no
+further than Example~\ref{ex:regular-mod-is-not-semisimple}: the regular
+\(\mathfrak{g}\)-module \(\mathcal{U}(\mathfrak{g})\) is never semisimple.
+Nevertheless, for simplicity -- or shall we say \emph{semisimplicity} -- we
+will focus exclusively on \emph{semisimple} \(\mathfrak{g}\)-modules. Our
+strategy is, once again, that of classifying simple modules. The regular
+\(\mathfrak{g}\)-module hides further unpleasant surprises, however: recall
+from Example~\ref{ex:regular-mod-is-not-weight-mod} that
\[
\bigoplus_\lambda \mathcal{U}(\mathfrak{g})_\lambda
= 0
@@ -38,12 +30,12 @@ there is still a diverse spectrum of counterexamples to
Corollary~\ref{thm:finite-dim-is-weight-mod} in the infinite-dimensional
setting. For instance, any \(\mathfrak{g}\)-module \(M\) whose restriction to
\(\mathfrak{h}\) is a free module satisfies \(M_\lambda = 0\) for all
-\(\lambda\) as in the previous example. These are called
-\emph{\(\mathfrak{h}\)-free \(\mathfrak{g}\)-modules}, and rank \(1\) simple
-\(\mathfrak{h}\)-free \(\mathfrak{sp}_{2 n}(K)\)-modules where first classified
-by Nilsson in \cite{nilsson}. Dimitar's construction of the so called
-\emph{exponential tensor \(\mathfrak{sl}_n(K)\)-modules} in \cite{dimitar-exp}
-is also an interesting source of counterexamples.
+\(\lambda\) as in Example~\ref{ex:regular-mod-is-not-weight-mod}. These are
+called \emph{\(\mathfrak{h}\)-free \(\mathfrak{g}\)-modules}, and rank \(1\)
+simple \(\mathfrak{h}\)-free \(\mathfrak{sp}_{2 n}(K)\)-modules where first
+classified by Nilsson in \cite{nilsson}. Dimitar's construction of the so
+called \emph{exponential tensor \(\mathfrak{sl}_n(K)\)-modules} in
+\cite{dimitar-exp} is also an interesting source of counterexamples.
Since the weight space decomposition was perhaps the single most instrumental
ingredient of our previous analysis, it is only natural to restrict ourselves
@@ -69,9 +61,10 @@ to the case it holds. This brings us to the following definition.
weight module for semisimple \(\mathfrak{g}\). In particular, if
\(\mathfrak{g}\) is finite-dimensional then the adjoint
\(\mathfrak{g}\)-module \(\mathfrak{g}\) is a weight module. More generally,
- a finite-dimensional Lie algebra \(\mathfrak{g}\) is reductive if, and only if
- the adjoint \(\mathfrak{g}\)-module \(\mathfrak{g}\) is a weight module, in
- which case its weight spaces are given by the root spaces of \(\mathfrak{g}\)
+ a finite-dimensional Lie algebra \(\mathfrak{g}\) is reductive if, and only
+ if the adjoint \(\mathfrak{g}\)-module \(\mathfrak{g}\) is a weight module,
+ in which case its weight spaces are given by the root spaces of
+ \(\mathfrak{g}\)
\end{example}
\begin{example}\label{ex:submod-is-weight-mod}
@@ -84,6 +77,8 @@ to the case it holds. This brings us to the following definition.
N\) for all \(\lambda \in \mathfrak{h}^*\).
\end{example}
+% TODO: Make this example shorter: it suffices to notice that M/N is the sum of
+% M_λ/N over λ
\begin{example}\label{ex:quotient-is-weight-mod}
Given a weight module \(M\), a submodule \(N \subset M\) and \(\lambda \in
\mathfrak{h}^*\), \(\left(\mfrac{M}{N}\right)_\lambda = \mfrac{M_\lambda}{N}
@@ -115,6 +110,26 @@ to the case it holds. This brings us to the following definition.
\left(\mfrac{M}{N}\right)_\lambda\) is surjective.
\end{example}
+\begin{example}\label{ex:tensor-prod-of-weight-is-weight}
+ Let \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\) be Lie algebras, \(M_1\) be a
+ weight \(\mathfrak{g}_1\)-module and \(M_2\) a weight
+ \(\mathfrak{g}_2\)-module. Recall from Example~\ref{ex:cartan-direct-sum}
+ that if \(\mathfrak{h}_i \subset \mathfrak{g}_i\) are Cartan subalgebras then
+ \(\mathfrak{h} = \mathfrak{h}_1 \oplus \mathfrak{h}_2\) is a Cartan
+ subalgebra of \(\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2\) with
+ \(\mathfrak{h}^* = \mathfrak{h}_1^* \oplus \mathfrak{h}_2^*\). In this
+ setting, one can readily check that \(M_1 \otimes M_2\) is a weight
+ \(\mathfrak{g}\)-module with
+ \[
+ (M_1 \otimes M_2)_{\lambda_1 + \lambda_2}
+ = (M_1)_{\lambda_1} \otimes (M_2)_{\lambda_2}
+ \]
+ for all \(\lambda_i \in \mathfrak{h}_i^*\) and \(\operatorname{supp} M_1
+ \otimes M_2 = \operatorname{supp} M_1 \oplus \operatorname{supp} M_2 = \{
+ \lambda_1 + \lambda_2 : \lambda_i \in \operatorname{supp} M_i \subset
+ \mathfrak{h}_i^*\}\).
+\end{example}
+
\begin{example}\label{thm:simple-weight-mod-is-tensor-prod}
Let \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
\mathfrak{s}_r\) be a reductive Lie algebra, where \(\mathfrak{z}\) is the
@@ -126,7 +141,8 @@ to the case it holds. This brings us to the following definition.
M \cong Z \otimes M_1 \otimes \cdots \otimes M_r
\]
where \(Z\) is a \(1\)-dimensional representation of \(\mathfrak{z}\) and
- \(M_i\) is a simple weight \(\mathfrak{s}_i\)-module.
+ \(M_i\) is a simple weight \(\mathfrak{s}_i\)-module. The modules \(Z\) and
+ \(M_i\) are uniquely determined up to isomorphism.
\end{example}
\begin{example}\label{ex:adjoint-action-in-universal-enveloping-is-weight}
@@ -150,106 +166,18 @@ to the case it holds. This brings us to the following definition.
contains \(\mathcal{U}(\mathfrak{h})\) and is therefore infinite-dimensional.
\end{example}
-We would like to stress that the weight spaces \(M_\lambda \subset M\) are
-\emph{not} \(\mathfrak{g}\)-submodules. Nevertheless, \(M_\lambda\) is a
-\(\mathfrak{h}\)-submodule. More generally, \(M_\lambda\) is a
-\(\mathcal{U}(\mathfrak{g})_0\)-submodule, where
-\(\mathcal{U}(\mathfrak{g})_0\) is the centralizer of \(\mathfrak{h}\) in
-\(\mathcal{U}(\mathfrak{g})\), and the multiplicities of simple
-\(\mathfrak{g}\)-modules in a semisimple weight \(\mathfrak{g}\)-module \(M\)
-are related to the multiplicities of simple
-\(\mathcal{U}(\mathfrak{g})_0\)-modules in \(M_\lambda\) via the following
-result.
-
-\begin{proposition}\label{thm:centralizer-multiplicity}
- Let \(\mathfrak{g}\) be a finite-dimensional reductive Lie algebra and \(M\)
- be a semisimple weight \(\mathfrak{g}\)-module. Then \(M_\lambda\) is a
- semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
- \operatorname{supp} M\). Moreover, if \(L\) is a simple weight
- \(\mathfrak{g}\)-module such that \(\lambda \in \operatorname{supp} L\) then
- \(L_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module and the
- multiplicity \(L\) in \(M\) coincides with the multiplicity of \(L_\lambda\)
- in \(M_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module.
-\end{proposition}
-
-\begin{proof}
- We begin by showing that \(L_\lambda\) is simple. Let \(N \subset L_\lambda\)
- be a nontrivial \(\mathcal{U}(\mathfrak{g})_0\)-submodule. We want to
- establish that \(N = L_\lambda\).
-
- If \(\mathcal{U}(\mathfrak{g})_\alpha\) denotes the root space of \(\alpha\)
- in \(\mathcal{U}(\mathfrak{g})\) under the adjoint action of \(\mathfrak{g}\)
- as in Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight},
- \(\alpha \in Q\), a simple calculation shows
- \(\mathcal{U}(\mathfrak{g})_\alpha \cdot N \subset L_{\lambda + \alpha}\).
- Since \(L\) is simple and \(N\) is nonzero, it follows from
- Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight} that
- \[
- L
- = \mathcal{U}(\mathfrak{g}) \cdot N
- = \bigoplus_{\alpha \in Q} \mathcal{U}(\mathfrak{g})_\alpha \cdot N
- \]
- and thus \(L_{\lambda + \alpha} = \mathcal{U}(\mathfrak{g})_\alpha \cdot N\).
- In particular, \(L_\lambda = \mathcal{U}(\mathfrak{g})_0 \cdot N \subset N\)
- and \(N = L_\lambda\).
-
- Now given a semisimple weight \(\mathfrak{g}\)-module \(M = \bigoplus_i M_i\)
- with \(M_i\) simple, it is clear \(M_\lambda = \bigoplus_i (M_i)_\lambda\).
- Each \((M_i)_\lambda\) is either \(0\) or a simple
- \(\mathcal{U}(\mathfrak{g})_0\)-module, so that \(M_\lambda\) is a semisimple
- \(\mathcal{U}(\mathfrak{g})_0\)-module. In addition, to see that the
- multiplicity of \(L\) in \(M\) coincides with the multiplicity of
- \(L_\lambda\) in \(M_\lambda\) it suffices to show that if \((M_i)_\lambda
- \cong (M_j)_\lambda\) are both nonzero then \(M_i \cong M_j\).
-
- If \(I(M_i) = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{g})_0}
- (M_i)_\lambda\), the inclusion of \(\mathcal{U}(\mathfrak{g})_0\)-modules
- \((M_i)_\lambda \to M_i\) induces a \(\mathfrak{g}\)-homomorphism
- \begin{align*}
- I(M_i) & \to M_i \\
- u \otimes m & \mapsto u \cdot m
- \end{align*}
-
- Since \(M_i\) is simple and \(\lambda \in \operatorname{supp} M_i\), \(M_i =
- \mathcal{U}(\mathfrak{g}) \cdot (M_i)_\lambda\). The homomorphism \(I(M_i)
- \to M_i\) is thus surjective. Similarly, if \(I(M_j) =
- \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{g})_0}
- (M_j)_\lambda\) then there is a natural surjective
- \(\mathfrak{g}\)-homomorphism \(I(M_j) \to M_j\). Now suppose there is an
- isomorphism of \(\mathcal{U}(\mathfrak{g})_0\)-modules \(f: (M_i)_\lambda
- \isoto (M_j)_\lambda\). Such an isomorphism induces an isomorphism of
- \(\mathfrak{g}\)-modules
- \begin{align*}
- \tilde f : I(M_i) & \isoto I(M_j) \\
- u \otimes m & \mapsto u \otimes f(m)
- \end{align*}
-
- By composing \(\tilde f\) with the projection \(I(M_j) \to M_j\) we get a
- surjective homomorphism \(I(M_i) \to M_j\). We claim \(\ker (I(M_i) \to M_i)
- = \ker (I(M_i) \to M_j)\). To see this, notice that \(\ker(I(M_i) \to M_i)\)
- coincides with the largest submodule \(Z(M_i) \subset I(M_i)\) contained in
- \(\bigoplus_{\alpha \ne 0} \mathcal{U}(\mathfrak{g})_\alpha
- \otimes_{\mathcal{U}(\mathfrak{g})_0} (M_i)_\lambda\). Indeed, a simple
- computation shows \(\ker (I(M_i) \to M_i) \cap (\mathcal{U}(\mathfrak{g})_0
- \otimes_{\mathcal{U}(\mathfrak{g})_0} (M_i)_\lambda) = 0\), which implies
- \(\ker(I(M_i) \to M_i) \subset Z(M_i)\). Since \(M_i\) is simple, \(\ker
- (I(M_i) \to M_i)\) is maximal and thus \(\ker(I(M_i) \to M_i) = Z(M_i)\). By
- the same token, \(\ker (I(M_j) \to M_j)\) is the largest submodule of
- \(I(M_j)\) contained in \(\bigoplus_{\alpha \ne 0}
- \mathcal{U}(\mathfrak{g})_\alpha \otimes_{\mathcal{U}(\mathfrak{g})_0}
- (M_j)_\lambda\) and therefore \(\ker(I(M_i) \to M_i) =
- \tilde{f}^{-1}(\ker(I(M_j) \to M_j)) = \ker(I(M_i) \to M_j)\).
-
- Hence there is an isomorphism \(\mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \isoto
- M_j\) satisfying
- \begin{center}
- \begin{tikzcd}
- I(M_i) \rar{\tilde f} \dar & I(M_j) \dar \\
- \mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \rar{\sim} & M_j
- \end{tikzcd}
- \end{center}
- and finally \(M_i \cong \mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \cong M_j\).
-\end{proof}
+\begin{note}
+ We should stress that the weight spaces \(M_\lambda \subset M\) of a given
+ weight \(\mathfrak{g}\)-module \(M\) are \emph{not}
+ \(\mathfrak{g}\)-submodules. Nevertheless, \(M_\lambda\) is a
+ \(\mathfrak{h}\)-submodule. More generally, \(M_\lambda\) is a
+ \(\mathcal{U}(\mathfrak{g})_0\)-submodule, where
+ \(\mathcal{U}(\mathfrak{g})_0\) is the centralizer of \(\mathfrak{h}\) in
+ \(\mathcal{U}(\mathfrak{g})\) -- which coincides with the weight space of \(0
+ \in \mathfrak{h}^*\) in the adjoint \(\mathfrak{g}\)-module
+ \(\mathcal{U}(\mathfrak{g})\), as seen in
+ Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight}.
+\end{note}
A particularly well behaved class of examples are the so called
\emph{bounded} modules.
@@ -262,6 +190,25 @@ A particularly well behaved class of examples are the so called
\{ \lambda \in \mathfrak{h}^* : \dim M_\lambda = \deg M \}\).
\end{definition}
+\begin{example}\label{ex:supp-ess-of-tensor-is-product}
+ Let \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\) be Lie algebras with Cartan
+ subalgebras \(\mathfrak{h}_i \subset \mathfrak{g}_i\) and take \(\mathfrak{g}
+ = \mathfrak{g}_1 \oplus \mathfrak{g}_2\). Given bounded
+ \(\mathfrak{g}_i\)-modules \(M_i\), it follows from
+ Example~\ref{ex:tensor-prod-of-weight-is-weight} that \(M_1 \otimes M_2\) is
+ a bounded \(\mathfrak{g}\)-module with \(\deg M_1 \otimes M_2 = \deg M_1
+ \cdot \deg M_2\) and
+ \[
+ \operatorname{supp}_{\operatorname{ess}} M_1 \otimes M_2
+ = \operatorname{supp}_{\operatorname{ess}} M_1 \oplus
+ \operatorname{supp}_{\operatorname{ess}} M_2
+ = \{
+ \lambda_1 + \lambda_2 : \lambda_i \in
+ \operatorname{supp}_{\operatorname{ess}} M_i \subset \mathfrak{h}_i^*
+ \}
+ \]
+\end{example}
+
\begin{example}\label{ex:laurent-polynomial-mod}
There is a natural action of \(\mathfrak{sl}_2(K)\) on the space \(K[x,
x^{-1}]\) of Laurent polynomials, given by the formulas in
@@ -301,36 +248,45 @@ precisely the entire \(Q\)-coset it inhabits -- i.e.
\(\operatorname{supp}_{\operatorname{ess}} K[x, x^{-1}] = 2 \mathbb{Z}\). This
isn't always the case. Nevertheless, in general we find\dots
-\begin{proposition}
- Let \(M\) be a simple infinite-dimensional bounded
- \(\mathfrak{g}\)-module. The essential support
- \(\operatorname{supp}_{\operatorname{ess}} M\) is Zariski-dense\footnote{Any
- choice of basis for $\mathfrak{h}^*$ induces a $K$-linear isomorphism
- $\mathfrak{h}^* \isoto K^n$. In particular, a choice of basis induces a
- unique topology in $\mathfrak{h}^*$ such that the map $\mathfrak{h}^* \to
- K^n$ is a homeomorphism onto $K^n$ with the Zariski topology. Any two basis
- induce the same topology in $\mathfrak{h}^*$, which we call \emph{the Zariski
- topology of $\mathfrak{h}^*$}.} in \(\mathfrak{h}^*\).
+\begin{proposition}\label{thm:ess-supp-is-zariski-dense}
+ Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra and \(M\)
+ be a simple infinite-dimensional bounded \(\mathfrak{g}\)-module. The
+ essential support \(\operatorname{supp}_{\operatorname{ess}} M\) is
+ Zariski-dense\footnote{Any choice of basis for $\mathfrak{h}^*$ induces a
+ $K$-linear isomorphism $\mathfrak{h}^* \isoto K^n$. In particular, a choice
+ of basis induces a unique topology in $\mathfrak{h}^*$ such that the map
+ $\mathfrak{h}^* \to K^n$ is a homeomorphism onto $K^n$ with the Zariski
+ topology. Any two basis induce the same topology in $\mathfrak{h}^*$, which
+ we call \emph{the Zariski topology of $\mathfrak{h}^*$}.} in
+ \(\mathfrak{h}^*\).
\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 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}} :
+3.5 of \cite{mathieu} for the case where \(\mathfrak{g} = \mathfrak{s}\) is a
+simple Lie algebra. The general case then follows from
+Example~\ref{thm:simple-weight-mod-is-tensor-prod},
+Example~\ref{ex:supp-ess-of-tensor-is-product} and the asserting that the
+product of Zariski-dense subsets in \(K^n\) and \(K^m\) is Zariski-dense in
+\(K^{n + m} = K^n \times K^m\).
+
+We now begin a systematic investigation of the problem of classifying the
+infinite-dimensional simple weight modules of a given Lie algebra
+\(\mathfrak{g}\). As in the previous chapter, let \(\mathfrak{g}\) be a
+finite-dimensional semisimple Lie algebra. As a first approximation of a
+solution to our problem, we consider the induction functors
+\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} :
\mathfrak{p}\text{-}\mathbf{Mod} \to \mathfrak{g}\text{-}\mathbf{Mod}\), where
-\(\mathfrak{p} \subset \mathfrak{g}\) is some subalgebra. These functors have
-already proved themselves a powerful tool for constructing modules in the
-previous chapters. Our first observation is that if \(\mathfrak{p} \subset
-\mathfrak{g}\) contains the Borel subalgebra \(\mathfrak{b}\) then
-\(\mathfrak{h}\) is a Cartan subalgebra of \(\mathfrak{p}\) and
-\((\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} M)_\lambda =
-\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{p})} M_\lambda\) for
-all \(\lambda \in \mathfrak{h}^*\). In particular,
+\(\mathfrak{p} \subset \mathfrak{g}\) is some subalgebra.
+
+% TODOO: Are you sure that these are indeed the weight spaces of the induced
+% module? Check this out?
+These functors have already proved themselves a powerful tool for constructing
+modules in the previous chapters. Our first observation is that if
+\(\mathfrak{p} \subset \mathfrak{g}\) contains the Borel subalgebra
+\(\mathfrak{b}\) then \(\mathfrak{h}\) is a Cartan subalgebra of
+\(\mathfrak{p}\) and \((\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}
+M)_\lambda = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{p})}
+M_\lambda\) for all \(\lambda \in \mathfrak{h}^*\). In particular,
\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}\) takes weight
\(\mathfrak{p}\)-modules to weight \(\mathfrak{g}\)-modules. This leads us to
the following definition.
@@ -340,6 +296,8 @@ the following definition.
if \(\mathfrak{b} \subset \mathfrak{p}\).
\end{definition}
+% TODOO: Why is the fact that p is not reductive relevant?? Why do we need to
+% look at the quotient by nil(p)??
Parabolic subalgebras thus give us a process for constructing weight
\(\mathfrak{g}\)-modules from modules of smaller (parabolic) subalgebras. Our
hope is that by iterating this process again and again we can get a large class
@@ -643,8 +601,9 @@ families}.
Our hope is that given a simple cuspidal module \(M\), we can somehow fit \(M\)
inside of a coherent \(\mathfrak{g}\)-family, such as in the case of \(K[x,
-x^{-1}]\) and \(\mathcal{M}\) from Example~\ref{ex:sl-laurent-family}. This
-leads us to the following definition.
+x^{-1}]\) and \(\mathcal{M}\) from Example~\ref{ex:sl-laurent-family}. In
+addition, we hope that such coherent families are somehow \emph{uniquely
+determined} by \(M\). This leads us to the following definition.
\begin{definition}\index{coherent family!coherent extension}
Given a bounded \(\mathfrak{g}\)-module \(M\) of degree \(d\), a
@@ -685,12 +644,14 @@ extension of \(M\).
% Mathieu's proof of this is somewhat profane, I don't think it's worth
% including it in here
+% TODO: Move this somewhere else? This holds in general for weight modules
+% whose suppert is contained in a single Q-coset
\begin{lemma}\label{thm:component-coh-family-has-finite-length}
Given a coherent family \(\mathcal{M}\) and \(\lambda \in \mathfrak{h}^*\),
\(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
\end{lemma}
-\begin{corollary}\index{coherent family!semisimplification}
+\begin{proposition}\index{coherent family!semisimplification}
Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a
unique semisimple coherent family \(\mathcal{M}^{\operatorname{ss}}\) of
degree \(d\) such that the composition series of
@@ -712,7 +673,7 @@ extension of \(M\).
\cong \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}}
\mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}
\]
-\end{corollary}
+\end{proposition}
\begin{proof}
The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear:
@@ -791,9 +752,11 @@ extension of \(M\).
\end{note}
The proof of Lemma~\ref{thm:component-coh-family-has-finite-length} is
-extremely technical and may be found in \cite{mathieu} -- see Lemma 3.3. As
-promised, if \(\mathcal{M}\) is a coherent extension of \(M\) then so is
-\(\mathcal{M}^{\operatorname{ss}}\).
+extremely technical and will not be included in here. It suffices to note that,
+as in Proposition~\ref{thm:ess-supp-is-zariski-dense}, the general case follows
+from the case where \(\mathfrak{g}\) is simple, which may be found in
+\cite{mathieu} -- see Lemma 3.3. As promised, if \(\mathcal{M}\) is a coherent
+extension of \(M\) then so is \(\mathcal{M}^{\operatorname{ss}}\).
\begin{proposition}
Let \(M\) be a simple bounded \(\mathfrak{g}\)-module and \(\mathcal{M}\)
@@ -828,8 +791,103 @@ itself and therefore\dots
\end{corollary}
These last results provide a partial answer to the question of existence of
-well behaved coherent extensions. A complementary question now is: which
-submodules of a \emph{nice} coherent family are cuspidal representations?
+well behaved coherent extensions. As for the uniqueness \(\mathcal{M}\) in
+Corollary~\ref{thm:bounded-is-submod-of-extension}, it suffices to show that
+the multiplicities of the simple weight \(\mathfrak{g}\)-modules in
+\(\mathcal{M}\) are uniquely determined by \(M\). These multiplicities may be
+computed via the following lemma.
+
+\begin{lemma}\label{thm:centralizer-multiplicity}
+ Let \(M\) be a semisimple weight \(\mathfrak{g}\)-module. Then \(M_\lambda\)
+ is a semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
+ \operatorname{supp} M\). Moreover, if \(L\) is a simple weight
+ \(\mathfrak{g}\)-module such that \(\lambda \in \operatorname{supp} L\) then
+ \(L_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module and the
+ multiplicity \(L\) in \(M\) coincides with the multiplicity of \(L_\lambda\)
+ in \(M_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module.
+\end{lemma}
+
+\begin{proof}
+ We begin by showing that \(L_\lambda\) is simple. Let \(N \subset L_\lambda\)
+ be a nontrivial \(\mathcal{U}(\mathfrak{g})_0\)-submodule. We want to
+ establish that \(N = L_\lambda\).
+
+ If \(\mathcal{U}(\mathfrak{g})_\alpha\) denotes the root space of \(\alpha\)
+ in \(\mathcal{U}(\mathfrak{g})\) under the adjoint action of \(\mathfrak{g}\)
+ as in Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight},
+ \(\alpha \in Q\), a simple calculation shows
+ \(\mathcal{U}(\mathfrak{g})_\alpha \cdot N \subset L_{\lambda + \alpha}\).
+ Since \(L\) is simple and \(N\) is nonzero, it follows from
+ Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight} that
+ \[
+ L
+ = \mathcal{U}(\mathfrak{g}) \cdot N
+ = \bigoplus_{\alpha \in Q} \mathcal{U}(\mathfrak{g})_\alpha \cdot N
+ \]
+ and thus \(L_{\lambda + \alpha} = \mathcal{U}(\mathfrak{g})_\alpha \cdot N\).
+ In particular, \(L_\lambda = \mathcal{U}(\mathfrak{g})_0 \cdot N \subset N\)
+ and \(N = L_\lambda\).
+
+ Now given a semisimple weight \(\mathfrak{g}\)-module \(M = \bigoplus_i M_i\)
+ with \(M_i\) simple, it is clear \(M_\lambda = \bigoplus_i (M_i)_\lambda\).
+ Each \((M_i)_\lambda\) is either \(0\) or a simple
+ \(\mathcal{U}(\mathfrak{g})_0\)-module, so that \(M_\lambda\) is a semisimple
+ \(\mathcal{U}(\mathfrak{g})_0\)-module. In addition, to see that the
+ multiplicity of \(L\) in \(M\) coincides with the multiplicity of
+ \(L_\lambda\) in \(M_\lambda\) it suffices to show that if \((M_i)_\lambda
+ \cong (M_j)_\lambda\) are both nonzero then \(M_i \cong M_j\).
+
+ If \(I(M_i) = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{g})_0}
+ (M_i)_\lambda\), the inclusion of \(\mathcal{U}(\mathfrak{g})_0\)-modules
+ \((M_i)_\lambda \to M_i\) induces a \(\mathfrak{g}\)-homomorphism
+ \begin{align*}
+ I(M_i) & \to M_i \\
+ u \otimes m & \mapsto u \cdot m
+ \end{align*}
+
+ Since \(M_i\) is simple and \(\lambda \in \operatorname{supp} M_i\), \(M_i =
+ \mathcal{U}(\mathfrak{g}) \cdot (M_i)_\lambda\). The homomorphism \(I(M_i)
+ \to M_i\) is thus surjective. Similarly, if \(I(M_j) =
+ \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{g})_0}
+ (M_j)_\lambda\) then there is a natural surjective
+ \(\mathfrak{g}\)-homomorphism \(I(M_j) \to M_j\). Now suppose there is an
+ isomorphism of \(\mathcal{U}(\mathfrak{g})_0\)-modules \(f: (M_i)_\lambda
+ \isoto (M_j)_\lambda\). Such an isomorphism induces an isomorphism of
+ \(\mathfrak{g}\)-modules
+ \begin{align*}
+ \tilde f : I(M_i) & \isoto I(M_j) \\
+ u \otimes m & \mapsto u \otimes f(m)
+ \end{align*}
+
+ By composing \(\tilde f\) with the projection \(I(M_j) \to M_j\) we get a
+ surjective homomorphism \(I(M_i) \to M_j\). We claim \(\ker (I(M_i) \to M_i)
+ = \ker (I(M_i) \to M_j)\). To see this, notice that \(\ker(I(M_i) \to M_i)\)
+ coincides with the largest submodule \(Z(M_i) \subset I(M_i)\) contained in
+ \(\bigoplus_{\alpha \ne 0} \mathcal{U}(\mathfrak{g})_\alpha
+ \otimes_{\mathcal{U}(\mathfrak{g})_0} (M_i)_\lambda\). Indeed, a simple
+ computation shows \(\ker (I(M_i) \to M_i) \cap (\mathcal{U}(\mathfrak{g})_0
+ \otimes_{\mathcal{U}(\mathfrak{g})_0} (M_i)_\lambda) = 0\), which implies
+ \(\ker(I(M_i) \to M_i) \subset Z(M_i)\). Since \(M_i\) is simple, \(\ker
+ (I(M_i) \to M_i)\) is maximal and thus \(\ker(I(M_i) \to M_i) = Z(M_i)\). By
+ the same token, \(\ker (I(M_j) \to M_j)\) is the largest submodule of
+ \(I(M_j)\) contained in \(\bigoplus_{\alpha \ne 0}
+ \mathcal{U}(\mathfrak{g})_\alpha \otimes_{\mathcal{U}(\mathfrak{g})_0}
+ (M_j)_\lambda\) and therefore \(\ker(I(M_i) \to M_i) =
+ \tilde{f}^{-1}(\ker(I(M_j) \to M_j)) = \ker(I(M_i) \to M_j)\).
+
+ Hence there is an isomorphism \(\mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \isoto
+ M_j\) satisfying
+ \begin{center}
+ \begin{tikzcd}
+ I(M_i) \rar{\tilde f} \dar & I(M_j) \dar \\
+ \mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \rar{\sim} & M_j
+ \end{tikzcd}
+ \end{center}
+ and finally \(M_i \cong \mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \cong M_j\).
+\end{proof}
+
+A complementary question now is: which submodules of a \emph{nice} coherent
+family are cuspidal representations?
\begin{proposition}[Mathieu]
Let \(\mathcal{M}\) be an irreducible coherent family of degree \(d\) and
@@ -842,6 +900,7 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
\end{enumerate}
\end{proposition}
+% TODO: Turn this first footnote into part of the proof
\begin{proof}
The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
from Corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
@@ -870,7 +929,7 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
In particular, \(M_\mu \ne 0\), so \(M_\mu = \mathcal{M}_\mu\). Now given any
simple \(\mathfrak{g}\)-module \(L\), it follows from
- Proposition~\ref{thm:centralizer-multiplicity} that the multiplicity of \(L\)
+ Lemma~\ref{thm:centralizer-multiplicity} that the multiplicity of \(L\)
in \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(L_\mu\) in
\(\mathcal{M}_\mu\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module -- which is,
of course, \(1\) if \(L \cong M\) and \(0\) otherwise. Hence
@@ -1476,7 +1535,7 @@ Lo and behold\dots
\(\mathfrak{g}\)-submodule of \(\mExt(M)\). Since the degree of \(M\) is the
same as the degree of \(\mExt(M)\), some of its weight spaces have maximal
dimension inside of \(\mExt(M)\). In particular, it follows from
- Proposition~\ref{thm:centralizer-multiplicity} that \(\mExt(M)_\lambda =
+ Lemma~\ref{thm:centralizer-multiplicity} that \(\mExt(M)_\lambda =
M_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module for some
\(\lambda \in \operatorname{supp} M\).