diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1,29 +1,30 @@
-\chapter{Irreducible Weight Modules}\label{ch:mathieu}
+\chapter{Simple Weight Modules}\label{ch:mathieu}
-In this chapter we will expand our results on finite-dimensional irreducible
-representations of semisimple Lie algebras by generalizing them on multiple
+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
-\(\mathfrak{g}\)-module which is not the direct sum of irreducible
-representations.
+\(\mathfrak{g}\)-module which is not the direct sum of simple
+\(\mathfrak{g}\)-modules.
-Nevertheless, completely reducible representations are a \emph{very} large
-class of \(\mathfrak{g}\)-modules, and understanding them can still give us a
-lot of information regarding our algebra and the category of its
-representations -- 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 representations. Our strategy is, once
-again, to classify the irreducible representations.
+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.
Secondly, and this is more important, we now consider
-\emph{infinite-dimensional} representations 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,
+\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,
\[
\bigoplus_\lambda \mathcal{U}(\mathfrak{g})_\lambda
= 0
@@ -33,13 +34,13 @@ and the weight space decomposition fails for \(\mathcal{U}(\mathfrak{g})\).
Indeed, our proof of the weight space decomposition in the finite-dimensional
case relied heavily in the simultaneous diagonalization of commuting operators
-in a finite-dimensional space. Even if we restrict ourselves to irreducible
-modules, there is still a diverse spectrum of counterexamples to
+in a finite-dimensional space. Even if we restrict ourselves to simple modules,
+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 representation \(V\) of \(\mathfrak{g}\) whose
-restriction to \(\mathfrak{h}\) is a free module satisfies \(V_\lambda = 0\)
-for all \(\lambda\) as in the previous example. These are called
-\emph{\(\mathfrak{h}\)-free representations}, and rank \(1\) irreducible
+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}
@@ -50,72 +51,72 @@ ingredient of our previous analysis, it is only natural to restrict ourselves
to the case it holds. This brings us to the following definition.
\begin{definition}
- A representation \(V\) of \(\mathfrak{g}\) is called a \emph{weight
- \(\mathfrak{g}\)-module} if \(V = \bigoplus_{\lambda \in \mathfrak{h}^*}
- V_\lambda\) and \(\dim V_\lambda < \infty\) for all \(\lambda \in
- \mathfrak{h}^*\). The \emph{support of \(V\)} is the set
- \(\operatorname{supp} V = \{\lambda \in \mathfrak{h}^* : V_\lambda \ne 0\}\).
+ A \(\mathfrak{g}\)-module \(M\) is called a \emph{weight
+ \(\mathfrak{g}\)-module} if \(M = \bigoplus_{\lambda \in \mathfrak{h}^*}
+ M_\lambda\) and \(\dim M_\lambda < \infty\) for all \(\lambda \in
+ \mathfrak{h}^*\). The \emph{support of \(M\)} is the set
+ \(\operatorname{supp} M = \{\lambda \in \mathfrak{h}^* : M_\lambda \ne 0\}\).
\end{definition}
\begin{example}
Corollary~\ref{thm:finite-dim-is-weight-mod} is equivalent to the fact that
- every finite-dimensional representation of a semisimple Lie algebra is a
- weight module. More generally, every finite-dimensional irreducible
- representation of a reductive Lie algebra is a weight module.
+ every finite-dimensional module of a semisimple Lie algebra is a weight
+ module. More generally, every finite-dimensional simple module of a reductive
+ Lie algebra is a weight module.
\end{example}
\begin{example}\label{ex:submod-is-weight-mod}
Proposition~\ref{thm:verma-is-weight-mod} and
Proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
- \(M(\lambda)\) and its maximal subrepresentation are both weight modules. In
+ \(M(\lambda)\) and its maximal submodule are both weight modules. In
fact, the proof of Proposition~\ref{thm:max-verma-submod-is-weight} is
- actually a proof of the fact that every subrepresentation \(W \subset V\) of
- a weight module \(V\) is a weight module, and \(W_\lambda = V_\lambda \cap
- W\) for all \(\lambda \in \mathfrak{h}^*\).
+ actually a proof of the fact that every submodule \(N \subset M\) of
+ a weight module \(M\) is a weight module, and \(N_\lambda = M_\lambda \cap
+ N\) for all \(\lambda \in \mathfrak{h}^*\).
\end{example}
\begin{example}\label{ex:quotient-is-weight-mod}
- Given a weight module \(V\), a submodule \(W \subset V\) and \(\lambda \in
- \mathfrak{h}^*\), \(\left(\mfrac{V}{W}\right)_\lambda = \mfrac{V_\lambda}{W}
- \cong \mfrac{V_\lambda}{W_\lambda}\). In particular,
+ Given a weight module \(M\), a submodule \(N \subset M\) and \(\lambda \in
+ \mathfrak{h}^*\), \(\left(\mfrac{M}{N}\right)_\lambda = \mfrac{M_\lambda}{N}
+ \cong \mfrac{M_\lambda}{N_\lambda}\). In particular,
\[
- \mfrac{V}{W}
- = \bigoplus_{\lambda \in \mathfrak{h}^*} \left(\mfrac{V}{W}\right)_\lambda
+ \mfrac{M}{N}
+ = \bigoplus_{\lambda \in \mathfrak{h}^*} \left(\mfrac{M}{N}\right)_\lambda
\]
- is a weight module. It is clear that \(\mfrac{V_\lambda}{W} \subset
- \left(\mfrac{V}{W}\right)_\lambda\). To see that \(\mfrac{V_\lambda}{W} =
- \left(\mfrac{V}{W}\right)_\lambda\), we remark that \(V_\lambda \cong
+ is a weight module. It is clear that \(\mfrac{M_\lambda}{N} \subset
+ \left(\mfrac{M}{N}\right)_\lambda\). To see that \(\mfrac{M_\lambda}{N} =
+ \left(\mfrac{M}{N}\right)_\lambda\), we remark that \(M_\lambda \cong
\mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} V\) as \(\mathfrak{h}\)-modules, where
+ \otimes_{\mathcal{U}(\mathfrak{h})} M\) as \(\mathfrak{h}\)-modules, where
\(I_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal
generated by the elements \(H - \lambda(H)\), \(H \in \mathfrak{h}\).
- Likewise \(\left(\mfrac{V}{W}\right)_\lambda \cong
+ Likewise \(\left(\mfrac{M}{N}\right)_\lambda \cong
\mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}\) and the diagram
+ \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}\) and the diagram
\begin{center}
\begin{tikzcd}
- V_\lambda \arrow{d} \arrow{r}{\pi} &
- \left(\mfrac{V}{W}\right)_\lambda \arrow{d} \\
+ M_\lambda \arrow{d} \arrow{r}{\pi} &
+ \left(\mfrac{M}{N}\right)_\lambda \arrow{d} \\
\mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} V
+ \otimes_{\mathcal{U}(\mathfrak{h})} M
\arrow[swap]{r}{\operatorname{id} \otimes \pi} &
\mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}
+ \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}
\end{tikzcd}
\end{center}
- commutes, so that the projection \(V_\lambda \to
- \left(\mfrac{V}{W}\right)_\lambda\) is surjective.
+ commutes, so that the projection \(M_\lambda \to
+ \left(\mfrac{M}{N}\right)_\lambda\) is surjective.
\end{example}
A particularly well behaved class of examples are the so called
\emph{admissible} weight modules.
\begin{definition}
- A weight \(\mathfrak{g}\)-module is called \emph{admissible} if \(\dim
- V_\lambda\) is bounded. The lowest upper bound \(d\) for \(\dim V_\lambda\)
- is called \emph{the degree of \(V\)}. The \emph{essential support} of \(V\)
- is the set \(\operatorname{supp}_{\operatorname{ess}} V = \{ \lambda \in
- \mathfrak{h}^* : \dim V_\lambda = d \}\).
+ A weight \(\mathfrak{g}\)-module \(M\) is called \emph{admissible} if \(\dim
+ M_\lambda\) is bounded. The lowest upper bound \(\deg M\) for \(\dim
+ M_\lambda\) is called \emph{the degree of \(M\)}. The \emph{essential
+ support} of \(M\) is the set \(\operatorname{supp}_{\operatorname{ess}} M =
+ \{ \lambda \in \mathfrak{h}^* : \dim M_\lambda = \deg M \}\).
\end{definition}
\begin{example}\label{ex:laurent-polynomial-mod}
@@ -126,13 +127,12 @@ A particularly well behaved class of examples are the so called
\notin 2 \mathbb{Z}\), so that \(K[x, x^{-1}] = \bigoplus_{k \in \mathbb{Z}}
K x^k\) is a degree \(1\) admissible weight \(\mathfrak{sl}_2(K)\)-module. It
follows from the remark at the end of Example~\ref{ex:submod-is-weight-mod}
- that any nonzero subrepresentation \(W \subset K[x, x^{-1}]\) must contain a
+ that any nonzero submodule \(N \subset K[x, x^{-1}]\) must contain a
monomial \(x^k\). But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} +
\frac{x^{-1}}{2}, x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} : K[x,
x^{-1}] \to K[x, x^{-1}]\) are both injective, this implies all other
- monomials can be found in \(W\) by successively applying \(f\) and \(e\).
- Hence \(W = K[x, x^{-1}]\) and \(K[x, x^{-1}]\) is an irreducible
- representation.
+ monomials can be found in \(N\) by successively applying \(f\) and \(e\).
+ Hence \(N = K[x, x^{-1}]\) and \(K[x, x^{-1}]\) is a simple module.
\begin{align}\label{eq:laurent-polynomials-cusp-mod}
e \cdot p
& = \left( x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} \right) p &
@@ -145,23 +145,23 @@ A particularly well behaved class of examples are the so called
Notice that the support of \(K[x, x^{-1}]\) is the trivial \(2
\mathbb{Z}\)-coset \(0 + 2 \mathbb{Z}\). This is representative of the general
-behavior in the following sense: if \(V\) is an irreducible weight
-\(\mathfrak{g}\)-module, since \(V[\lambda] = \bigoplus_{\alpha \in Q}
-V_{\lambda + \alpha}\) is stable under the action of \(\mathfrak{g}\) for all
-\(\lambda \in \mathfrak{h}^*\), \(\bigoplus_{\alpha \in Q} V_{\lambda +
-\alpha}\) is either \(0\) or all of \(V\). In other words, the support of an
-irreducible weight module is always contained in a single \(Q\)-coset.
+behavior in the following sense: if \(M\) is a simple weight
+\(\mathfrak{g}\)-module, since \(M[\lambda] = \bigoplus_{\alpha \in Q}
+M_{\lambda + \alpha}\) is stable under the action of \(\mathfrak{g}\) for all
+\(\lambda \in \mathfrak{h}^*\), \(\bigoplus_{\alpha \in Q} M_{\lambda +
+\alpha}\) is either \(0\) or all of \(M\). In other words, the support of a
+simple weight module is always contained in a single \(Q\)-coset.
However, the behavior of \(K[x, x^{-1}]\) deviates from that of an arbitrary
-admissible representation in the sense its essential support is precisely the
-entire \(Q\)-coset it inhabits -- i.e.
+admissible \(\mathfrak{g}\)-module in the sense its essential support is
+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 \(V\) be an infinite-dimensional admissible representation of
- \(\mathfrak{g}\). The essential support
- \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense\footnote{Any
+ Let \(M\) be an infinite-dimensional admissible
+ \(\mathfrak{g}\)-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
@@ -175,19 +175,19 @@ This proof was deemed too technical to be included in here, but see Proposition
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
-irreducible weight \(\mathfrak{g}\)-modules for some fixed reductive Lie
-algebra \(\mathfrak{g}\).
+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 \(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} :
\mathfrak{p}\text{-}\mathbf{Mod} \to \mathfrak{g}\text{-}\mathbf{Mod}\), where
\(\mathfrak{p} \subset \mathfrak{g}\) is some subalgebra. These functors have
-already proved themselves a powerful tool for constructing representations in
-the previous chapters. Our first observation is that if \(\mathfrak{p} \subset
+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
-\((\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V)_\lambda =
-\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{p})} V_\lambda\) for
+\((\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} M)_\lambda =
+\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{p})} M_\lambda\) 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
@@ -199,42 +199,43 @@ the following definition.
\end{definition}
Parabolic subalgebras thus give us a process for constructing weight
-\(\mathfrak{g}\)-modules from representations of smaller (parabolic)
-subalgebras. Our hope is that by iterating this process again and again we can
-get a large class of irreducible weight \(\mathfrak{g}\)-modules. However,
-there is a small catch: a parabolic subalgebra \(\mathfrak{p} \subset
-\mathfrak{g}\) needs not to be reductive. We can get around this limitation by
-modding out by \(\mathfrak{u} = \mathfrak{nil}(\mathfrak{p})\) and noticing
-that \(\mathfrak{u}\) acts trivially in any weight \(\mathfrak{p}\)-module
-\(V\). By applying the universal property of quotients we can see that \(V\)
-has the natural structure of a representation of
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\), which is always a reductive algebra.
+\(\mathfrak{g}\)-modules from modules of smaller (parabolic) subalgebras. Our
+hope is that by iterating this process again and again we can get a large class
+of simple weight \(\mathfrak{g}\)-modules. However, there is a small catch: a
+parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) needs not to be
+reductive. We can get around this limitation by modding out by
+\(\mathfrak{nil}(\mathfrak{p})\) and noticing that
+\(\mathfrak{nil}(\mathfrak{p})\) acts trivially in any weight
+\(\mathfrak{p}\)-module \(M\). By applying the universal property of quotients
+we can see that \(M\) has the natural structure of a
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module, which is always
+a reductive algebra.
\begin{center}
\begin{tikzcd}
- \mathfrak{p} \rar \dar & \mathfrak{gl}(V) \\
- \mfrac{\mathfrak{p}}{\mathfrak{u}} \arrow[dotted]{ur} &
+ \mathfrak{p} \rar \dar & \mathfrak{gl}(M) \\
+ \mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})} \arrow[dotted]{ur} &
\end{tikzcd}
\end{center}
-Let \(\mathfrak{p}\) be a parabolic subalgebra and \(V\) be an irreducible
+Let \(\mathfrak{p}\) be a parabolic subalgebra and \(M\) be a simple
weight \(\mathfrak{p}\)-module. We should point out that while
-\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is a weight
-\(\mathfrak{g}\)-module, it isn't necessarily irreducible. Nevertheless, we can
-use it to produce an irreducible weight \(\mathfrak{g}\)-module via a
+\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} M\) is a weight
+\(\mathfrak{g}\)-module, it isn't necessarily simple. Nevertheless, we can
+use it to produce a simple weight \(\mathfrak{g}\)-module via a
construction very similar to that of Verma modules.
\begin{definition}
- The module \(M_{\mathfrak{p}}(V) =
- \operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is called \emph{a
+ Given any \(\mathfrak{p}\)-module \(M\), the module \(M_{\mathfrak{p}}(M) =
+ \operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} M\) is called \emph{a
generalized Verma module}.
\end{definition}
\begin{proposition}\label{thm:generalized-verma-has-simple-quotient}
- Given an irreducible \(\mathfrak{p}\)-module \(V\), the generalized Verma
- module \(M_{\mathfrak{p}}(V)\) has a unique maximal subrepresentation
- \(N_{\mathfrak{p}}(V)\) and a unique irreducible quotient
- \(L_{\mathfrak{p}}(V) = \mfrac{M_{\mathfrak{p}}(V)}{N_{\mathfrak{p}}(V)}\).
- The irreducible quotient \(L_{\mathfrak{p}}(V)\) is a weight module.
+ Given a simple \(\mathfrak{p}\)-module \(M\), the generalized Verma module
+ \(M_{\mathfrak{p}}(M)\) has a unique maximal \(\mathfrak{p}\)-submodule
+ \(N_{\mathfrak{p}}(M)\) and a unique irreducible quotient
+ \(L_{\mathfrak{p}}(M) = \mfrac{M_{\mathfrak{p}}(M)}{N_{\mathfrak{p}}(M)}\).
+ The irreducible quotient \(L_{\mathfrak{p}}(M)\) is a weight module.
\end{proposition}
The proof of Proposition~\ref{thm:generalized-verma-has-simple-quotient} is
@@ -242,63 +243,62 @@ entirely analogous to that of Proposition~\ref{thm:max-verma-submod-is-weight}.
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
- subalgebra \(\mathfrak{p} \subsetneq \mathfrak{g}\) and some
- \(\mathfrak{p}\)-module \(V\). An \emph{irreducible cuspidal
- \(\mathfrak{g}\)-module} is an irreducible representation of \(\mathfrak{g}\)
- which is \emph{not} parabolic induced.
+ A \(\mathfrak{g}\)-module is called \emph{parabolic induced} if it is
+ isomorphic to \(L_{\mathfrak{p}}(M)\) for some proper parabolic subalgebra
+ \(\mathfrak{p} \subsetneq \mathfrak{g}\) and some \(\mathfrak{p}\)-module
+ \(M\). An \emph{simple cuspidal \(\mathfrak{g}\)-module} is a simple
+ \(\mathfrak{g}\)-module which is \emph{not} parabolic induced.
\end{definition}
-Since every weight \(\mathfrak{p}\)-module \(V\) is an
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\)-module, it makes sense to call \(V\)
-\emph{cuspidal} if it is a cuspidal representation of
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\). The first breakthrough regarding our
-classification problem was given by Fernando in his now infamous paper
-\citetitle{fernando} \cite{fernando}, where he proved that every irreducible
-weight \(\mathfrak{g}\)-module is parabolic induced. In other words\dots
+Since every weight \(\mathfrak{p}\)-module \(M\) is an
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module, it makes sense to call \(M\)
+\emph{cuspidal} if it is a cuspidal
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module. The first breakthrough regarding
+our classification problem was given by Fernando in his now infamous paper
+\citetitle{fernando} \cite{fernando}, where he proved that every simple weight
+\(\mathfrak{g}\)-module is parabolic induced. 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 \(V\).
+ Any simple weight \(\mathfrak{g}\)-module is isomorphic to
+ \(L_{\mathfrak{p}}(M)\) for some parabolic subalgebra \(\mathfrak{p} \subset
+ \mathfrak{g}\) and some simple cuspidal \(\mathfrak{p}\)-module \(M\).
\end{theorem}
-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. Nevertheless, this
-relationship is well understood. Namely, Fernando himself established\dots
+We should point out that the relationship between simple weight
+\(\mathfrak{g}\)-modules and pairs \((\mathfrak{p}, M)\) -- where
+\(\mathfrak{p}\) is some parabolic subalgebra and \(M\) is a simple cuspidal
+\(\mathfrak{p}\)-module -- is not one-to-one. Nevertheless, this relationship
+is well understood. Namely, Fernando himself established\dots
\begin{proposition}[Fernando]
Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\), there
exists a basis \(\Sigma\) for \(\Delta\) such that \(\Sigma \subset
\Delta_{\mathfrak{p}} \subset \Delta\), where \(\Delta_{\mathfrak{p}}\)
denotes the set of roots of \(\mathfrak{p}\). 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 \(L_{\mathfrak{p}}(V) \cong
- L_{\mathfrak{p}'}(W)\) if, and only if \(\mathfrak{p}' =
- \mathfrak{p}^\sigma\) and \(V \cong \sigma W\) for some\footnote{Here
+ \subset \mathfrak{g}\) is another parabolic subalgebra, \(M\) is a simple
+ cuspidal \(\mathfrak{p}\)-module and \(N\) is a simple
+ cuspidal \(\mathfrak{p}'\)-module then \(L_{\mathfrak{p}}(M) \cong
+ L_{\mathfrak{p}'}(N)\) if, and only if \(\mathfrak{p}' =
+ \mathfrak{p}^\sigma\) and \(M \cong \sigma N\) for some\footnote{Here
$\mathfrak{p}^\sigma$ denotes the image of $\mathfrak{p}$ under the
automorphism of $\sigma : \mathfrak{g} \to \mathfrak{g}$ given by the
- canonical action of $\mathcal{W}$ on $\mathfrak{g}$ and $\sigma W$ is the
+ canonical action of $\mathcal{W}$ on $\mathfrak{g}$ and $\sigma N$ is the
$\mathfrak{p}$-module given by composing the map $\mathfrak{p}' \to
- \mathfrak{gl}(W)$ with the restriction $\sigma\!\restriction_{\mathfrak{p}} :
- \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in \mathcal{W}_V\), where
+ \mathfrak{gl}(N)$ with the restriction $\sigma\!\restriction_{\mathfrak{p}} :
+ \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in \mathcal{W}_M\), where
\[
- \mathcal{W}_V
+ \mathcal{W}_M
= \langle
- \sigma_\beta : \beta \in \Sigma, H_\beta + \mathfrak{u}
- \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{u}}
- \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ V
+ \sigma_\beta : \beta \in \Sigma, H_\beta + \mathfrak{nil}(\mathfrak{p})
+ \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}
+ \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ M
\rangle
\subset \mathcal{W}
\]
\end{proposition}
\begin{note}
- The definition of the subgroup \(\mathcal{W}_V \subset \mathcal{W}\) is
+ The definition of the subgroup \(\mathcal{W}_M \subset \mathcal{W}\) is
independent of the choice of basis \(\Sigma\).
\end{note}
@@ -306,42 +306,40 @@ As a first consequence of Fernando's Theorem, we provide two alternative
characterizations of cuspidal modules.
\begin{corollary}[Fernando]\label{thm:cuspidal-mod-equivs}
- Let \(V\) be an irreducible weight \(\mathfrak{g}\)-module. The following
+ Let \(M\) be a simple weight \(\mathfrak{g}\)-module. The following
conditions are equivalent.
\begin{enumerate}
- \item \(V\) is cuspidal
- \item \(F_\alpha\) acts injectively in \(V\) for all
+ \item \(M\) is cuspidal
+ \item \(F_\alpha\) acts injectively on \(M\) for all
\(\alpha \in \Delta\) -- this is what is usually referred
- to as a \emph{dense} representation in the literature
- \item The support of \(V\) is precisely one \(Q\)-coset -- this is
- what is usually referred to as a \emph{torsion-free} representation in the
+ to as a \emph{dense} module in the literature
+ \item The support of \(M\) is precisely one \(Q\)-coset -- this is
+ what is usually referred to as a \emph{torsion-free} module in the
literature
\end{enumerate}
\end{corollary}
\begin{example}
As noted in Example~\ref{ex:laurent-polynomial-mod}, the element \(f \in
- \mathfrak{sl}_2(K)\) acts injectively in the space of Laurent polynomials.
- Hence \(K[x, x^{-1}]\) is a cuspidal representation of
- \(\mathfrak{sl}_2(K)\).
+ \mathfrak{sl}_2(K)\) acts injectively on the space of Laurent polynomials.
+ Hence \(K[x, x^{-1}]\) is a cuspidal \(\mathfrak{sl}_2(K)\)-module.
\end{example}
-Having reduced our classification problem to that of classifying irreducible
-cuspidal representations, we are now faced the daunting task of actually
-classifying them. Historically, this was first achieved by Olivier Mathieu in
-the early 2000's in his paper \citetitle{mathieu} \cite{mathieu}. To do so,
-Mathieu introduced new tools which have since proved themselves remarkably
-useful throughout the field, known as\dots
+Having reduced our classification problem to that of classifying simple
+cuspidal modules, we are now faced the daunting task of actually classifying
+them. Historically, this was first achieved by Olivier Mathieu in the early
+2000's in his paper \citetitle{mathieu} \cite{mathieu}. To do so, Mathieu
+introduced new tools which have since proved themselves remarkably useful
+throughout the field, known as\dots
\section{Coherent Families}
-We begin our analysis with a simple question: how to do we go about constructing
-cuspidal representations? Specifically, given a cuspidal
-\(\mathfrak{g}\)-module, how can we use it to produce new cuspidal
-representations? To answer this question, we look back at the single example of
-a cuspidal representation we have encountered so far: the
-\(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\) of Laurent polynomials -- i.e.
-Example~\ref{ex:laurent-polynomial-mod}.
+We begin our analysis with a simple question: how to do we go about
+constructing cuspidal modules? Specifically, given a cuspidal
+\(\mathfrak{g}\)-module, how can we use it to produce new cuspidal modules? To
+answer this question, we look back at the single example of a cuspidal module
+we have encountered so far: the \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\)
+of Laurent polynomials -- i.e. Example~\ref{ex:laurent-polynomial-mod}.
Our first observation is that \(\mathfrak{sl}_2(K)\) acts on \(K[x, x^{-1}]\)
via differential operators. In other words, the action map
@@ -371,8 +369,8 @@ automorphisms of \(\operatorname{Diff}(K[x, x^{-1}])\). For example, given
\frac{\lambda}{2} x^{-1}
\end{align*}
and consider the module \(\varphi_\lambda K[x, x^{-1}] = K[x, x^{-1}]\) where
-some operator \(L \in \operatorname{Diff}(K[x, x^{-1}])\) acts as
-\(\varphi_\lambda(L)\).
+some operator \(P \in \operatorname{Diff}(K[x, x^{-1}])\) acts as
+\(\varphi_\lambda(P)\).
By composing the action map \(\operatorname{Diff}(K[x, x^{-1}]) \to
\operatorname{End}(\varphi_\lambda K[x, x^{-1}])\) with the homomorphism of
@@ -414,11 +412,11 @@ Hence \(\varphi_\lambda K[x, x^{-1}]\) is a degree \(1\) admissible
x^{-1}] = \frac{\lambda}{2} + 2 \mathbb{Z}\). One can also quickly check that
if \(\lambda \notin 1 + 2 \mathbb{Z}\) then \(e\) and \(f\) act injectively in
\(\varphi_\lambda K[x, x^{-1}]\), so that \(\varphi_\lambda K[x, x^{-1}]\) is
-irreducible. In particular, if \(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with
+simple. In particular, if \(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with
\(\lambda \notin \mu + 2 \mathbb{Z}\) then \(\varphi_\lambda K[x, x^{-1}]\) and
-\(\varphi_\mu K[x, x^{-1}]\) are non-isomorphic irreducible cuspidal
+\(\varphi_\mu K[x, x^{-1}]\) are non-isomorphic simple cuspidal
\(\mathfrak{sl}_2(K)\)-modules, since their supports differ. These cuspidal
-representations can be ``glued together'' in a \emph{monstrous concoction} by
+modules can be ``glued together'' in a \emph{monstrous concoction} by
summing over \(\lambda \in K\), as in
\[
\mathcal{M}
@@ -439,10 +437,10 @@ concoction''.
On a perhaps less derogatory note, \(\mathcal{M}\) also deserves to be called
\emph{a family}. This is because \(\mathcal{M}\) consists of lots of smaller
-cuspidal representations which fit together inside of it in a \emph{coherent}
-fashion. Mathieu's ingenious breakthrough was the realization that
-\(\mathcal{M}\) is a particular example of a more general pattern, which he
-named \emph{coherent families}.
+cuspidal modules which fit together inside of it in a \emph{coherent} fashion.
+Mathieu's ingenious breakthrough was the realization that \(\mathcal{M}\) is a
+particular example of a more general pattern, which he named \emph{coherent
+families}.
\begin{definition}
A \emph{coherent family \(\mathcal{M}\) of degree \(d\)} is a weight
@@ -495,47 +493,46 @@ named \emph{coherent families}.
\end{example}
\begin{note}
- We would lie to stress that coherent families have proven themselves useful
+ We would like to stress that coherent families have proven themselves useful
for problems other than the classification of cuspidal
\(\mathfrak{g}\)-modules. For instance, Nilsson's classification of rank 1
- \(\mathfrak{h}\)-free representations of \(\mathfrak{sp}_{2 n}(K)\) is based
- on the notion of coherent families and the so called \emph{weighting
- functor}.
+ \(\mathfrak{h}\)-free \(\mathfrak{sp}_{2 n}(K)\)-modules is based on the
+ notion of coherent families and the so called \emph{weighting functor}.
\end{note}
-Our hope is that given an irreducible cuspidal representation \(V\), we can
-somehow fit \(V\) 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.
+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.
\begin{definition}
- Given an admissible \(\mathfrak{g}\)-module \(V\) of degree \(d\), a
- \emph{coherent extension \(\mathcal{M}\) of \(V\)} is a coherent family
- \(\mathcal{M}\) of degree \(d\) that contains \(V\) as a subquotient.
+ Given an admissible \(\mathfrak{g}\)-module \(M\) of degree \(d\), a
+ \emph{coherent extension \(\mathcal{M}\) of \(M\)} is a coherent family
+ \(\mathcal{M}\) of degree \(d\) that contains \(M\) as a subquotient.
\end{definition}
-Our goal is now showing that every admissible representation has a coherent
-extension. The idea then is to classify coherent extensions, and classify which
-submodules of a given coherent extension are actually irreducible cuspidal
-representations. If every admissible \(\mathfrak{g}\)-module fits inside a
-coherent extension, this would lead to classification of all irreducible
-cuspidal representations, which we now know is the key for the solution of our
-classification problem. However, there are some complications to this scheme.
+Our goal is now showing that every admissible module has a coherent extension.
+The idea then is to classify coherent families, and classify which submodules
+of a given coherent family are actually simple cuspidal modules. If every
+admissible \(\mathfrak{g}\)-module fits inside a coherent extension, this would
+lead to classification of all simple cuspidal \(\mathfrak{g}\)-modules, which
+we now know is the key for the solution of our classification problem. However,
+there are some complications to this scheme.
Leaving aside the question of existence for a second, we should point out that
coherent families turn out to be rather complicated on their own. In fact they
are too complicated to classify in general. Ideally, we would like to find
\emph{nice} coherent extensions -- ones we can actually classify. For instance,
-we may search for \emph{simple} coherent extensions, which are defined as
+we may search for \emph{irreducible} coherent extensions, which are defined as
follows.
\begin{definition}
- A coherent family \(\mathcal{M}\) is called \emph{simple} if it contains no
- proper coherent subfamilies -- i.e. \(\mathcal{M}\) is a simple object in the
- full subcategory of coherent families. Equivalently, we call \(\mathcal{M}\)
- simple if \(\mathcal{M}_\lambda\) is a simple
- \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
- \mathfrak{h}^*\).
+ A coherent family \(\mathcal{M}\) is called \emph{irreducible} if it contains
+ no proper coherent subfamilies -- i.e. \(\mathcal{M}\) is a simple object in
+ the full subcategory of \(\mathfrak{g}\text{-}\mathbf{Mod}\) consisting of
+ coherent families. Equivalently, we call \(\mathcal{M}\) irreducible if
+ \(\mathcal{M}_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module
+ for some \(\lambda \in \mathfrak{h}^*\).
\end{definition}
Another natural candidate for the role of ``nice extensions'' are the completely
@@ -543,15 +540,14 @@ reducible coherent families -- i.e. families which are completely reducible 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 \(V\)
-to a completely reducible coherent extension of \(V\).
+of modules.} of a coherent family}, which takes a coherent extension of \(M\)
+to a completely reducible coherent extension of \(M\).
% Mathieu's proof of this is somewhat profane, I don't think it's worth
% including it in here
\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
- \(\mathcal{U}(\mathfrak{g})\)-module.
+ \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
\end{lemma}
\begin{corollary}
@@ -564,7 +560,7 @@ to a completely reducible coherent extension of \(V\).
Namely, if \(\lambda \in \mathfrak{h}^*\) and \(0 = \mathcal{M}_{\lambda 0}
\subset \mathcal{M}_{\lambda 1} \subset \cdots \subset \mathcal{M}_{\lambda
- n_\lambda} = \mathcal{M}[\lambda]\) is a composition series\footnote{Notice
+ r_\lambda} = \mathcal{M}[\lambda]\) is a composition series\footnote{Notice
that $\mathcal{M}[\lambda] = \mathcal{M}[\mu]$ for any $\mu \in \lambda + Q$.
Hence the sum $\bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}
\bigoplus_i \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}$ is
@@ -642,38 +638,39 @@ to a completely reducible coherent extension of \(V\).
\begin{note}
Although we have provided an explicit construction of
\(\mathcal{M}^{\operatorname{ss}}\) in terms of \(\mathcal{M}\), we should
- point out this construction is not functorial. First, given an intertwiner \(T
- : \mathcal{M} \to \mathcal{N}\) between coherent families, it is unclear what
- \(T^{\operatorname{ss}} : \mathcal{M}^{\operatorname{ss}} \to
- \mathcal{N}^{\operatorname{ss}}\) is supposed to be. Secondly, and this is
- more relevant, our construction depends on the choice of composition series
- \(0 = \mathcal{M}_{\lambda 0} \subset \cdots \subset \mathcal{M}_{\lambda
- n_\lambda} = \mathcal{M}[\lambda]\). While different choices of composition
- series yield isomorphic results, there is no canonical isomorphism.
- In addition, there is no canonical choice of composition series.
+ point out this construction is not functorial. First, given a
+ \(\mathfrak{g}\)-homomorphism \(f : \mathcal{M} \to \mathcal{N}\) between
+ coherent families, it is unclear what \(f^{\operatorname{ss}} :
+ \mathcal{M}^{\operatorname{ss}} \to \mathcal{N}^{\operatorname{ss}}\) is
+ supposed to be. Secondly, and this is more relevant, our construction depends
+ on the choice of composition series \(0 = \mathcal{M}_{\lambda 0} \subset
+ \cdots \subset \mathcal{M}_{\lambda r_\lambda} = \mathcal{M}[\lambda]\).
+ While different choices of composition series yield isomorphic results, there
+ is no canonical isomorphism. In addition, there is no canonical choice of
+ composition series.
\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 \(V\) then so is
+promised, if \(\mathcal{M}\) is a coherent extension of \(M\) then so is
\(\mathcal{M}^{\operatorname{ss}}\).
\begin{proposition}
- Let \(V\) be an irreducible admissible \(\mathfrak{g}\)-module and
- \(\mathcal{M}\) be a coherent extension of \(V\). Then
- \(\mathcal{M}^{\operatorname{ss}}\) is a coherent extension of \(V\), and
- \(V\) is in fact a subrepresentation of \(\mathcal{M}^{\operatorname{ss}}\).
+ Let \(M\) be a simple admissible \(\mathfrak{g}\)-module and \(\mathcal{M}\)
+ be a coherent extension of \(M\). Then \(\mathcal{M}^{\operatorname{ss}}\) is
+ a coherent extension of \(M\), and \(M\) is in fact a submodule of
+ \(\mathcal{M}^{\operatorname{ss}}\).
\end{proposition}
\begin{proof}
- Since \(V\) is irreducible, its support is contained in a single \(Q\)-coset.
- This implies that \(V\) is a subquotient of \(\mathcal{M}[\lambda]\) for any
- \(\lambda \in \operatorname{supp} V\). If we fix some composition series \(0
- = \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
- \mathcal{M}[\lambda]\) of \(\mathcal{M}[\lambda]\) with \(V \cong
+ Since \(M\) is simple, its support is contained in a single \(Q\)-coset.
+ This implies that \(M\) is a subquotient of \(\mathcal{M}[\lambda]\) for any
+ \(\lambda \in \operatorname{supp} M\). If we fix some composition series \(0
+ = \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_r =
+ \mathcal{M}[\lambda]\) of \(\mathcal{M}[\lambda]\) with \(M \cong
\mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}\), there is a natural inclusion
\[
- V
+ M
\isoto \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i}
\to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j}
\cong \mathcal{M}^{\operatorname{ss}}[\lambda]
@@ -685,9 +682,9 @@ completely reducible coherent extension \(\mathcal{M}\) is \(\mathcal{M}\)
itself and therefore\dots
\begin{corollary}\label{thm:admissible-is-submod-of-extension}
- Let \(V\) be an irreducible admissible \(\mathfrak{g}\)-module and
- \(\mathcal{M}\) be a completely reducible coherent extension of \(V\). Then
- \(V\) is contained in \(\mathcal{M}\).
+ Let \(M\) be a simple admissible \(\mathfrak{g}\)-module and \(\mathcal{M}\)
+ be a completely reducible coherent extension of \(M\). Then \(M\) is
+ contained in \(\mathcal{M}\).
\end{corollary}
These last results provide a partial answer to the question of existence of
@@ -695,21 +692,22 @@ 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 \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
- \(V_\lambda\) 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 irreducible representation \(W\) of
- \(\mathfrak{g}\) coincides with the multiplicity of \(W_\lambda\) in
- \(V_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module, for any \(\lambda
- \in \operatorname{supp} V\).
+ 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 \(\lambda \in
+ \mathfrak{h}^*\), 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
+ multiplicity of \(L_\lambda\) in \(M_\lambda\) as a
+ \(\mathcal{U}(\mathfrak{g})_0\)-module, for any \(\lambda \in
+ \operatorname{supp} M\).
\end{proposition}
\begin{corollary}[Mathieu]
- Let \(\mathcal{M}\) be a simple coherent family of degree \(d\) and
+ Let \(\mathcal{M}\) be an irreducible coherent family of degree \(d\) and
\(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent.
\begin{enumerate}
- \item \(\mathcal{M}[\lambda]\) is irreducible.
+ \item \(\mathcal{M}[\lambda]\) is simple.
\item \(F_\alpha\!\restriction_{\mathcal{M}[\lambda]}\) is injective for
all \(\alpha \in \Delta\).
\item \(\mathcal{M}[\lambda]\) is cuspidal.
@@ -722,33 +720,33 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
corollary that \strong{(iii)} implies \strong{(ii)}. All it is left is to show
\strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
Corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
- $\mathcal{M}[\lambda]$ may not be irreducible for some $\lambda$ satisfying
+ $\mathcal{M}[\lambda]$ may not be simple for some $\lambda$ satisfying
\strong{(ii)}. We will show this is never the case.}.
- Suppose \(F_\alpha\) acts injectively in the subrepresentation
+ Suppose \(F_\alpha\) acts injectively on the submodule
\(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since
\(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains
- an infinite-dimensional irreducible \(\mathfrak{g}\)-submodule \(V\).
- Moreover, again by Corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(V\)
- is a cuspidal representation, and its degree is bounded by \(d\). We want to
- show \(\mathcal{M}[\lambda] = V\).
+ an infinite-dimensional simple \(\mathfrak{g}\)-submodule \(M\). Moreover,
+ again by Corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(M\) is a
+ cuspidal module, and its degree is bounded by \(d\). We want to show
+ \(\mathcal{M}[\lambda] = M\).
We claim the set \(U = \{\mu \in \mathfrak{h}^* : \mathcal{M}_\mu \ \text{is
a simple $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is Zariski-open. If we
suppose this is the case for a moment or two, it follows from the fact that
- \(\mathcal{M}\) is simple and \(\operatorname{supp}_{\operatorname{ess}} V\)
- is Zariski-dense that \(U \cap \operatorname{supp}_{\operatorname{ess}} V\)
+ \(\mathcal{M}\) is simple and \(\operatorname{supp}_{\operatorname{ess}} M\)
+ is Zariski-dense that \(U \cap \operatorname{supp}_{\operatorname{ess}} M\)
is non-empty. In other words, there is some \(\mu \in \mathfrak{h}^*\) such
that \(\mathcal{M}_\mu\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module
- and \(\dim V_\mu = \deg V\).
+ and \(\dim M_\mu = \deg M\).
- In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any
- irreducible \(\mathfrak{g}\)-module \(W\), it follows from
- Proposition~\ref{thm:centralizer-multiplicity} that the multiplicity of \(W\)
- in \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) in
+ 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 \(L\)
+ 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 \(W \cong V\) and \(0\) otherwise. Hence
- \(\mathcal{M}[\lambda] = V\) and \(\mathcal{M}[\lambda]\) is cuspidal.
+ of course, \(1\) if \(L \cong M\) and \(0\) otherwise. Hence
+ \(\mathcal{M}[\lambda] = M\) and \(\mathcal{M}[\lambda]\) is cuspidal.
\end{proof}
Once more, the proof of Proposition~\ref{thm:centralizer-multiplicity} wasn't
@@ -809,10 +807,10 @@ deemed informative enough to be included in here, but see the proof of Lemma
\(\mathcal{U}(\mathfrak{g})_0 \to \operatorname{End}(\mathcal{M}_\lambda)\)
is surjective. In particular, if \(\operatorname{rank} B_\lambda = d^2\) then
\(\mathcal{M}_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module,
- for if \(V \subset \mathcal{M}_\lambda\) is invariant under the action of
- \(\mathcal{U}(\mathfrak{g})_0\) then \(V\) is invariant under any
+ for if \(M \subset \mathcal{M}_\lambda\) is invariant under the action of
+ \(\mathcal{U}(\mathfrak{g})_0\) then \(M\) is invariant under any
\(K\)-linear operator \(\mathcal{M}_\lambda \to \mathcal{M}_\lambda\), so
- that \(W = 0\) or \(W = \mathcal{M}_\lambda\).
+ that \(M = 0\) or \(M = \mathcal{M}_\lambda\).
On the other hand, if \(\mathcal{M}_\lambda\) is simple then by Burnside's
Theorem on matrix algebras the map \(\mathcal{U}(\mathfrak{g})_0 \to
@@ -827,53 +825,53 @@ deemed informative enough to be included in here, but see the proof of Lemma
Zariski-open. First, notice that
\[
U =
- \bigcup_{\substack{W \subset \mathcal{U}(\mathfrak{g})_0 \\ \dim W = d^2}}
- U_W,
+ \bigcup_{\substack{V \subset \mathcal{U}(\mathfrak{g})_0 \\ \dim V = d}}
+ U_V,
\]
- where \(U_W = \{\lambda \in \mathcal{U}(\mathfrak{g})_0 : \operatorname{rank}
- B_\lambda\!\restriction_W = d^2 \}\). Here \(W\) ranges over all
- \(d\)-dimensional subspaces of \(\mathcal{U}(\mathfrak{g})_0\) -- \(W\) is
+ where \(U_V = \{\lambda \in \mathcal{U}(\mathfrak{g})_0 : \operatorname{rank}
+ B_\lambda\!\restriction_V = d^2 \}\). Here \(V\) ranges over all
+ \(d\)-dimensional subspaces of \(\mathcal{U}(\mathfrak{g})_0\) -- \(V\) is
not necessarily a \(\mathcal{U}(\mathfrak{g})_0\)-submodule.
Indeed, if \(\operatorname{rank} B_\lambda = d^2\) it follows from the
subjectivity of the map \(\mathcal{U}(\mathfrak{g})_0 \to
- \operatorname{End}(\mathcal{M}_\lambda)\) that there is some \(W \subset
- \mathcal{U}(\mathfrak{g})_0\) with \(\dim W = d^2\) such that the restriction
- \(W \to \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. The
+ \operatorname{End}(\mathcal{M}_\lambda)\) that there is some \(V \subset
+ \mathcal{U}(\mathfrak{g})_0\) with \(\dim V = d\) such that the restriction
+ \(V \to \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. The
commutativity of
\begin{center}
\begin{tikzcd}
- W \arrow{r} \arrow{d} & W^* \\
+ V \arrow{r} \arrow{d} & V^* \\
\operatorname{End}(\mathcal{M}_\lambda) \arrow{r}{\sim} &
\operatorname{End}(\mathcal{M}_\lambda)^* \arrow{u}
\end{tikzcd}
\end{center}
- then implies \(\operatorname{rank} B_\lambda\!\restriction_W = d^2\). In
- other words, \(U \subset \bigcup_W U_W\).
+ then implies \(\operatorname{rank} B_\lambda\!\restriction_V = d^2\). In
+ other words, \(U \subset \bigcup_V U_V\).
- Likewise, if \(\operatorname{rank} B_\lambda\!\restriction_W = d^2\) for some
- \(W\), then the commutativity of
+ Likewise, if \(\operatorname{rank} B_\lambda\!\restriction_V = d^2\) for some
+ \(V\), then the commutativity of
\begin{center}
\begin{tikzcd}
- W \arrow{r} \arrow{d} & W^* \\
+ V \arrow{r} \arrow{d} & V^* \\
\mathcal{U}(\mathfrak{g})_0 \arrow{r} &
\mathcal{U}(\mathfrak{g})_0^* \arrow{u}
\end{tikzcd}
\end{center}
implies \(\operatorname{rank} B_\lambda \ge d^2\), which goes to show
- \(\bigcup_W U_W \subset U\).
+ \(\bigcup_V U_V \subset U\).
- Given \(\lambda \in U_W\), the surjectivity of \(W \to
- \operatorname{End}(\mathcal{M}_\lambda)\) and the fact that \(\dim W <
- \infty\) imply \(W \to W^*\) is invertible. Since \(\mathcal{M}\) is a
+ Given \(\lambda \in U_V\), the surjectivity of \(V \to
+ \operatorname{End}(\mathcal{M}_\lambda)\) and the fact that \(\dim V <
+ \infty\) imply \(V \to V^*\) is invertible. Since \(\mathcal{M}\) is a
coherent family, \(B_\lambda\) depends polynomially in \(\lambda\). Hence so
- does the induced maps \(W \to W^*\). In particular, there is some Zariski
- neighborhood \(V\) of \(\lambda\) such that the map \(W \to W^*\) induced by
- \(B_\mu\!\restriction_W\) is invertible for all \(\mu \in V\).
+ does the induced maps \(V \to V^*\). In particular, there is some Zariski
+ neighborhood \(U'\) of \(\lambda\) such that the map \(V \to V^*\) induced by
+ \(B_\mu\!\restriction_V\) is invertible for all \(\mu \in U'\).
- But the surjectivity of the map induced by \(B_\mu\!\restriction_W\) implies
- \(\operatorname{rank} B_\mu = d^2\), so \(\mu \in U_W\) and therefore \(V
- \subset U_W\). This implies \(U_W\) is open for all \(W\). Finally, \(U\) is
+ But the surjectivity of the map induced by \(B_\mu\!\restriction_V\) implies
+ \(\operatorname{rank} B_\mu = d^2\), so \(\mu \in U_V\) and therefore \(U'
+ \subset U_V\). This implies \(U_V\) is open for all \(V\). Finally, \(U\) is
the union of Zariski-open subsets and is therefore open. We are done.
\end{proof}
@@ -882,15 +880,15 @@ coherent extensions, which will be the focus of our next section.
\section{Localizations \& the Existence of Coherent Extensions}
-Let \(V\) be an irreducible admissible \(\mathfrak{g}\)-module of degree \(d\).
-Our goal is to prove that \(V\) has a (unique) simple completely reducible
-coherent extension \(\mathcal{M}\). Since \(V\) is irreducible, we know \(V
-\subset \mathcal{M}[\lambda]\) for any \(\lambda \in \operatorname{supp} V\).
+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
-\(\operatorname{supp}_{\operatorname{ess}} V\) may not be all of \(\lambda + Q
+\(\operatorname{supp}_{\operatorname{ess}} M\) may not be all of \(\lambda + Q
= \operatorname{supp}_{\operatorname{ess}} \mathcal{M}[\lambda]\), so we may
-find \(V \subsetneq \mathcal{M}[\lambda]\). In fact, we may find
-\(\operatorname{supp} V \subsetneq \lambda + Q\).
+find \(M \subsetneq \mathcal{M}[\lambda]\). In fact, we may find
+\(\operatorname{supp} M \subsetneq \lambda + Q\).
This wasn't an issue an Example~\ref{ex:laurent-polynomial-mod} because we
verified that the action of \(f \in \mathfrak{sl}_2(K)\) on \(K[x, x^{-1}]\) is
@@ -914,15 +912,15 @@ inverse of the action of \(f\) on \(K[x, x^{-1}]\).
\end{center}
In the general case, the action of some \(F_\alpha \in \mathfrak{g}\) with
-\(\alpha \in \Delta\) in \(V\) may not be injective. In fact, we have seen that
-the action of \(F_\alpha\) is injective for all \(\alpha \in \Delta\) if, and
-only if \(V\) is cuspidal. Nevertheless, we could intuitively \emph{make it
+\(\alpha \in \Delta\) in \(M\) may not be injective. In fact, we have seen that
+the action of \(F_\alpha\) is injective for all \(\alpha \in \Delta^+\) if, and
+only if \(M\) is cuspidal. Nevertheless, we could intuitively \emph{make it
injective} by formally inverting the elements \(F_\alpha \in
\mathcal{U}(\mathfrak{g})\). This would allow us to obtain nonzero vectors in
-\(V_\mu\) for all \(\mu \in \lambda + Q\) by successively applying elements of
-\(\{F_\alpha^{\pm 1}\}_{\alpha \in \Delta}\) to a nonzero weight vector \(v \in
-V_\lambda\). Moreover, if the actions of the \(F_\alpha\) were to be
-invertible, we would find that all \(V_\mu\) are \(d\)-dimensional for \(\mu
+\(M_\mu\) for all \(\mu \in \lambda + Q\) by successively applying elements of
+\(\{F_\alpha^{\pm 1}\}_{\alpha \in \Delta}\) to a nonzero weight vector \(m \in
+M_\lambda\). Moreover, if the actions of the \(F_\alpha\) were to be
+invertible, we would find that all \(M_\mu\) are \(d\)-dimensional for \(\mu
\in \lambda + Q\).
In a commutative domain, this can be achieved by tensoring our module by the
@@ -986,13 +984,13 @@ instance\dots
\begin{lemma}
Let \(S \subset A\) be a multiplicative subset generated by finitely many
locally \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\)
- such that for each \(a \in A\) there exists \(n > 0\) such that
- \(\operatorname{ad}(s)^n a = [s, [s, \cdots [s, a]]\cdots] = 0\). Then \(S\)
+ such that for each \(a \in A\) there exists \(r > 0\) such that
+ \(\operatorname{ad}(s)^r a = [s, [s, \cdots [s, a]]\cdots] = 0\). Then \(S\)
satisfies Ore's localization condition.
\end{lemma}
In our case, we are more interested in formally inverting the action of
-\(F_\alpha\) on \(V\) than in inverting \(F_\alpha\) itself. To that end, we
+\(F_\alpha\) on \(M\) than in inverting \(F_\alpha\) itself. To that end, we
introduce one further construction, known as \emph{the localization of a
module}.
@@ -1024,30 +1022,30 @@ the injectivity of the localization map, as shown in\dots
\begin{lemma}
Let \(S \subset A\) be a multiplicative subset satisfying Ore's localization
- condition and \(M\) be an \(A\)-module. If \(S\) acts injectively in \(M\)
+ condition and \(M\) be an \(A\)-module. If \(S\) acts injectively on \(M\)
then the localization map \(M \to S^{-1} M\) is injective. In particular, if
\(S\) has no zero divisors then \(A\) is a subalgebra of \(S^{-1} A\).
\end{lemma}
Again, in our case we are interested in inverting the actions of the
-\(F_\alpha\) on \(V\). However, for us to be able to translate between all
+\(F_\alpha\) on \(M\). However, for us to be able to translate between all
weight spaces associated with elements of \(\lambda + Q\), \(\lambda \in
-\operatorname{supp} V\), we only need to invert the \(F_\alpha\)'s for
+\operatorname{supp} M\), we only need to invert the \(F_\alpha\)'s for
\(\alpha\) in some subset of \(\Delta\) which spans all of \(Q = \mathbb{Z}
\Delta\). In other words, it suffices to invert \(F_\beta\) for all \(\beta\)
in some basis \(\Sigma\) for \(\Delta\). We can choose such a basis to be
well-behaved. For example, we can show\dots
\begin{lemma}\label{thm:nice-basis-for-inversion}
- Let \(V\) be an irreducible infinite-dimensional admissible
+ Let \(M\) be a simple infinite-dimensional admissible
\(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
- \beta_n\}\) for \(\Delta\) such that the elements \(F_{\beta_i}\) all act
- injectively on \(V\) and satisfy \([F_{\beta_i}, F_{\beta_j}] = 0\).
+ \beta_r\}\) for \(\Delta\) such that the elements \(F_{\beta_i}\) all act
+ injectively on \(M\) and satisfy \([F_{\beta_i}, F_{\beta_j}] = 0\).
\end{lemma}
\begin{note}
The basis \(\Sigma\) in Lemma~\ref{thm:nice-basis-for-inversion} may very
- well depend on the representation \(V\)! This is another obstruction to the
+ well depend on the representation \(M\)! This is another obstruction to the
functoriality of our constructions.
\end{note}
@@ -1062,33 +1060,33 @@ to be included in here. See Lemma 4.4 of \cite{mathieu} for a full proof. Since
multiplicative subset generated by the \(F_\beta\)'s. The \(K\)-algebra
\(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta \in \Sigma}^{-1}
\mathcal{U}(\mathfrak{g})\) is well defined. Moreover, if we denote by
- \(\Sigma^{-1} V\) the localization of \(V\) by \((F_\beta)_{\beta \in
- \Sigma}\), the localization map \(V \to \Sigma^{-1} V\) is injective.
+ \(\Sigma^{-1} M\) the localization of \(M\) by \((F_\beta)_{\beta \in
+ \Sigma}\), the localization map \(M \to \Sigma^{-1} M\) is injective.
\end{corollary}
From now on let \(\Sigma\) be some fixed basis for \(\Delta\) satisfying the
hypothesis of Lemma~\ref{thm:nice-basis-for-inversion}. We now show that
-\(\Sigma^{-1} V\) is a weight module whose support is an entire \(Q\)-coset.
+\(\Sigma^{-1} M\) is a weight \(\mathfrak{g}\)-module whose support is an
+entire \(Q\)-coset.
\begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
- The restriction of the localization \(\Sigma^{-1} V\) is an admissible
+ The restriction of the localization \(\Sigma^{-1} M\) is an admissible
\(\mathfrak{g}\)-module of degree \(d\) with \(\operatorname{supp}
- \Sigma^{-1} V = Q + \operatorname{supp} V\) and \(\dim \Sigma^{-1} V_\lambda
- = d\) for all \(\lambda \in \operatorname{supp} \Sigma^{-1} V\).
+ \Sigma^{-1} M = Q + \operatorname{supp} M\) and \(\dim \Sigma^{-1} M_\lambda
+ = d\) for all \(\lambda \in \operatorname{supp} \Sigma^{-1} M\).
\end{proposition}
\begin{proof}
Fix some \(\beta \in \Sigma\). We begin by showing that \(F_\beta\) and
- \(F_\beta^{-1}\) map the weight space \(\Sigma^{-1} V_\lambda\) to
- \(\Sigma^{-1} V_{\lambda - \beta}\) and \(\Sigma^{-1} V_{\lambda + \beta}\),
- respectively. Indeed, given \(v \in V_\lambda\) and \(H \in \mathfrak{h}\) we
+ \(F_\beta^{-1}\) map the weight space \(\Sigma^{-1} M_\lambda\) to
+ \(\Sigma^{-1} M_{\lambda - \beta}\) and \(\Sigma^{-1} M_{\lambda + \beta}\),
+ respectively. Indeed, given \(m \in M_\lambda\) and \(H \in \mathfrak{h}\) we
have
\[
- H F_\beta v
- = ([H, F_\beta] + F_\beta H)v
- = F_\beta (-\beta(H) + H) v
- = F_\beta (\lambda - \beta)(H) \cdot v
- = (\lambda - \beta)(H) \cdot F_\beta v
+ H \cdot (F_\beta \cdot m)
+ = ([H, F_\beta] + F_\beta H) \cdot m
+ = F_\beta (-\beta(H) + H) \cdot m
+ = (\lambda - \beta)(H) F_\beta \cdot m
\]
On the other hand,
@@ -1101,44 +1099,45 @@ hypothesis of Lemma~\ref{thm:nice-basis-for-inversion}. We now show that
\]
so that \([H, F_\beta^{-1}] = \beta(H) \cdot F_\beta^{-1}\) and therefore
\[
- H F_\beta^{-1} v
- = ([H, F_\beta^{-1}] + F_\beta^{-1} H) v
- = F_\beta^{-1} (\beta(H) + H) v
- = (\lambda + \beta)(H) \cdot F_\beta^{-1} v
+ H \cdot (F_\beta^{-1} \cdot m)
+ = ([H, F_\beta^{-1}] + F_\beta^{-1} H) \cdot m
+ = F_\beta^{-1} (\beta(H) + H) \cdot m
+ = (\lambda + \beta)(H) F_\beta^{-1} \cdot m
\]
- From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(\Sigma^{-1}
- V_{\lambda \pm \beta}\) follows our first conclusion: since \(V\) is a weight
- module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
- s^{-1} \otimes v\) for \(s \in (F_\beta)_{\beta \in \Sigma}\) and \(v \in
- V\), we can see that \(\Sigma^{-1} V = \bigoplus_\lambda \Sigma^{-1}
- V_\lambda\). Furthermore, since the action of each \(F_\beta\) on
- \(\Sigma^{-1} V\) is bijective and \(\Sigma\) is a basis for \(Q\) we obtain
- \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\).
+ From the fact that \(F_\beta^{\pm 1}\) maps \(M_\lambda\) to \(\Sigma^{-1}
+ M_{\lambda \pm \beta}\) follows our first conclusion: since \(M\) is a weight
+ module and every element of \(\Sigma^{-1} M\) has the form \(s^{-1} \cdot m =
+ s^{-1} \otimes m\) for \(s \in (F_\beta)_{\beta \in \Sigma}\) and \(m \in
+ M\), we can see that \(\Sigma^{-1} M = \bigoplus_\lambda \Sigma^{-1}
+ M_\lambda\). Furthermore, since the action of each \(F_\beta\) on
+ \(\Sigma^{-1} M\) is bijective and \(\Sigma\) is a basis for \(Q\) we obtain
+ \(\operatorname{supp} \Sigma^{-1} M = Q + \operatorname{supp} M\).
Again, because of the bijectivity of the \(F_\beta\)'s, to see that \(\dim
- \Sigma^{-1} V_\lambda = d\) for all \(\lambda \in \operatorname{supp}
- \Sigma^{-1} V\) it suffices to show that \(\dim \Sigma^{-1} V_\lambda = d\)
- for some \(\lambda \in \operatorname{supp} \Sigma^{-1} V\). We may take
- \(\lambda \in \operatorname{supp} V\) with \(\dim V_\lambda = d\). For any
- finite-dimensional subspace \(W \subset \Sigma^{-1} V_\lambda\) we can find
- \(s \in (F_\beta)_{\beta \in \Sigma}\) such that \(s W \subset V\). If \(s =
- F_{\beta_{i_1}} \cdots F_{\beta_{i_n}}\), it is clear \(s W \subset
- V_{\lambda - \beta_{i_1} - \cdots - \beta_{i_n}}\), so \(\dim W = \dim sW \le
- d\). This holds for all finite-dimensional \(W \subset \Sigma^{-1}
- V_\lambda\), so \(\dim \Sigma^{-1} V_\lambda \le d\). It then follows from
- the fact that \(V_\lambda \subset \Sigma^{-1} V_\lambda\) that \(V_\lambda =
- \Sigma^{-1} V_\lambda\) and therefore \(\dim \Sigma^{-1} V_\lambda = d\).
+ \Sigma^{-1} M_\lambda = d\) for all \(\lambda \in \operatorname{supp}
+ \Sigma^{-1} M\) it suffices to show that \(\dim \Sigma^{-1} M_\lambda = d\)
+ for some \(\lambda \in \operatorname{supp} \Sigma^{-1} M\). We may take
+ \(\lambda \in \operatorname{supp} M\) with \(\dim M_\lambda = d\). For any
+ finite-dimensional subspace \(V \subset \Sigma^{-1} M_\lambda\) we can find
+ \(s \in (F_\beta)_{\beta \in \Sigma}\) such that \(s \cdot V \subset M\). If
+ \(s = F_{\beta_{i_1}} \cdots F_{\beta_{i_r}}\), it is clear \(s \cdot V
+ \subset M_{\lambda - \beta_{i_1} - \cdots - \beta_{i_r}}\), so \(\dim V =
+ \dim s \cdot V \le d\). This holds for all finite-dimensional \(V \subset
+ \Sigma^{-1} M_\lambda\), so \(\dim \Sigma^{-1} M_\lambda \le d\). It then
+ follows from the fact that \(M_\lambda \subset \Sigma^{-1} M_\lambda\) that
+ \(M_\lambda = \Sigma^{-1} M_\lambda\) and therefore \(\dim \Sigma^{-1}
+ M_\lambda = d\).
\end{proof}
-We now have a good candidate for a coherent extension of \(V\), but
-\(\Sigma^{-1} V\) is still not a coherent extension since its support is
+We now have a good candidate for a coherent extension of \(M\), but
+\(\Sigma^{-1} M\) is still not a coherent extension since its support is
contained in a single \(Q\)-coset. In particular, \(\operatorname{supp}
-\Sigma^{-1} V \ne \mathfrak{h}^*\) and \(\Sigma^{-1} V\) is not a coherent
+\Sigma^{-1} M \ne \mathfrak{h}^*\) and \(\Sigma^{-1} M\) is not a coherent
family. To obtain a coherent family we thus need somehow extend \(\Sigma^{-1}
-V\). To that end, we will attempt to replicate the construction of the coherent
+M\). To that end, we will attempt to replicate the construction of the coherent
extension of the \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\). Specifically,
-the idea is that if twist \(\Sigma^{-1} V\) by an automorphism which shifts its
+the idea is that if twist \(\Sigma^{-1} M\) by an automorphism which shifts its
support by some \(\lambda \in \mathfrak{h}^*\), we can construct a coherent
family by summing these modules over \(\lambda\) as in
Example~\ref{ex:sl-laurent-family}.
@@ -1149,27 +1148,27 @@ automorphisms \(\varphi_\lambda : \operatorname{Diff}(K[x, x^{-1}]) \to
\operatorname{Diff}(K[x, x^{-1}])\) and restricting the results to
\(\mathcal{U}(\mathfrak{sl}_2(K))\) via the map
\(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x, x^{-1}])\), but
-this approach is inflexible since not every representation of
-\(\mathfrak{sl}_2(K)\) factors through \(\operatorname{Diff}(K[x, x^{-1}])\).
-Nevertheless, we could just as well twist \(K[x, x^{-1}]\) by automorphisms of
+this approach is inflexible since not every \(\mathfrak{sl}_2(K)\)-module
+factors through \(\operatorname{Diff}(K[x, x^{-1}])\). Nevertheless, we could
+just as well twist \(K[x, x^{-1}]\) by automorphisms of
\(\mathcal{U}(\mathfrak{sl}_2(K))_f\) directly -- where
\(\mathcal{U}(\mathfrak{sl}_2(K))_f = (f)^{-1} \mathcal{U}(\mathfrak{g})\) is
the localization of \(\mathcal{U}(\mathfrak{sl}_2(K))\) by the multiplicative
subset generated by \(f\).
In general, we may twist the \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
-\(\Sigma^{-1} V\) by automorphisms of \(\Sigma^{-1}
+\(\Sigma^{-1} M\) by automorphisms of \(\Sigma^{-1}
\mathcal{U}(\mathfrak{g})\). For \(\lambda = \beta \in \Sigma\) the map
\begin{align*}
\theta_\beta : \Sigma^{-1} \mathcal{U}(\mathfrak{g}) & \to
\Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
- r & \mapsto F_\beta r F_\beta^{-1}
+ u & \mapsto F_\beta u F_\beta^{-1}
\end{align*}
is a natural candidate for such a twisting automorphism. Indeed, we will soon
-see that \((\theta_\beta \Sigma^{-1} V)_\lambda = \Sigma^{-1} V_{\lambda +
+see that \((\theta_\beta \Sigma^{-1} M)_\lambda = \Sigma^{-1} M_{\lambda +
\beta}\). However, this is hardly useful to us, since \(\beta \in Q\) and
-therefore \(\beta + \operatorname{supp} \Sigma^{-1} V = \operatorname{supp}
-\Sigma^{-1} V\). If we want to expand the support of \(\Sigma^{-1} V\) we will
+therefore \(\beta + \operatorname{supp} \Sigma^{-1} M = \operatorname{supp}
+\Sigma^{-1} M\). If we want to expand the support of \(\Sigma^{-1} M\) we will
have to twist by automorphisms that shift its support by \(\lambda \in
\mathfrak{h}^*\) lying \emph{outside} of \(Q\).
@@ -1183,26 +1182,26 @@ Explicitly\dots
\mathcal{U}(\mathfrak{g}) \to \Sigma^{-1}
\mathcal{U}(\mathfrak{g})\}_{\lambda \in \mathfrak{h}^*}\) such that
\begin{enumerate}
- \item \(\theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r) = F_{\beta_1}^{k_1}
- \cdots F_{\beta_n}^{k_n} r F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{-
- k_1}\) for all \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and \(k_1,
- \ldots, k_n \in \mathbb{Z}\).
+ \item \(\theta_{k_1 \beta_1 + \cdots + k_r \beta_r}(u) = F_{\beta_1}^{k_1}
+ \cdots F_{\beta_r}^{k_r} u F_{\beta_r}^{- k_r} \cdots F_{\beta_1}^{-
+ k_1}\) for all \(u \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and \(k_1,
+ \ldots, k_r \in \mathbb{Z}\).
- \item For each \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) the map
+ \item For each \(u \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) the map
\begin{align*}
\mathfrak{h}^* & \to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
- \lambda & \mapsto \theta_\lambda(r)
+ \lambda & \mapsto \theta_\lambda(u)
\end{align*}
is polynomial.
- \item If \(\lambda, \mu \in \mathfrak{h}^*\), \(M\) is a \(\Sigma^{-1}
+ \item If \(\lambda, \mu \in \mathfrak{h}^*\), \(N\) is a \(\Sigma^{-1}
\mathcal{U}(\mathfrak{g})\)-module whose restriction to
\(\mathcal{U}(\mathfrak{g})\) is a weight \(\mathfrak{g}\)-module and
- \(\theta_\lambda M\) is the \(\Sigma^{-1}
- \mathcal{U}(\mathfrak{g})\)-module \(M\) twisted by the automorphism
- \(\theta_\lambda\) then \(M_\mu = (\theta_\lambda M)_{\mu + \lambda}\).
- In particular, \(\operatorname{supp} \theta_\lambda M = \lambda +
- \operatorname{supp} M\).
+ \(\theta_\lambda N\) is the \(\Sigma^{-1}
+ \mathcal{U}(\mathfrak{g})\)-module \(N\) twisted by the automorphism
+ \(\theta_\lambda\) then \(N_\mu = (\theta_\lambda N)_{\mu + \lambda}\).
+ In particular, \(\operatorname{supp} \theta_\lambda N = \lambda +
+ \operatorname{supp} N\).
\end{enumerate}
\end{proposition}
@@ -1210,67 +1209,67 @@ Explicitly\dots
Since the elements \(F_\beta\), \(\beta \in \Sigma\) commute with one
another, the endomorphisms
\begin{align*}
- \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}
+ \theta_{k_1 \beta_1 + \cdots + k_r \beta_r}
: \Sigma^{-1} \mathcal{U}(\mathfrak{g}) &
\to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
- r & \mapsto
- F_{\beta_1}^{k_1} \cdots F_{\beta_n}^{k_n}
- r
- F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{- k_1}
+ u & \mapsto
+ F_{\beta_1}^{k_1} \cdots F_{\beta_r}^{k_r}
+ u
+ F_{\beta_1}^{- k_r} \cdots F_{\beta_r}^{- k_1}
\end{align*}
- are well defined for all \(k_1, \ldots, k_n \in \mathbb{Z}\).
-
- Fix some \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\). For any \(s \in
- (F_\beta)_{\beta \in \Sigma}\) and \(k > 0\) we have \(s^k r = \binom{k}{0}
- \operatorname{ad}(s)^0 r s^{k - 0} + \cdots + \binom{k}{k}
- \operatorname{ad}(s)^k r s^{k - k}\). Now if we take \(m\) such
- \(\operatorname{ad}(F_\beta)^{m + 1} r = 0\) for all \(\beta \in \Sigma\) we
- find
+ are well defined for all \(k_1, \ldots, k_r \in \mathbb{Z}\).
+
+ Fix some \(u \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\). For any \(s \in
+ (F_\beta)_{\beta \in \Sigma}\) and \(k > 0\) we have \(s^k u = \binom{k}{0}
+ \operatorname{ad}(s)^0 u s^{k - 0} + \cdots + \binom{k}{k}
+ \operatorname{ad}(s)^k u s^{k - k}\). Now if we take \(\ell\) such
+ \(\operatorname{ad}(F_\beta)^{\ell + 1} u = 0\) for all \(\beta \in \Sigma\)
+ we find
\[
- \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r)
- = \sum_{i_1, \ldots, i_n = 1, \ldots, m}
- \binom{k_1}{i_1} \cdots \binom{k_n}{i_n}
+ \theta_{k_1 \beta_1 + \cdots + k_r \beta_r}(u)
+ = \sum_{i_1, \ldots, i_r = 1, \ldots, \ell}
+ \binom{k_1}{i_1} \cdots \binom{k_r}{i_r}
\operatorname{ad}(F_{\beta_1})^{i_1} \cdots
- \operatorname{ad}(F_{\beta_n})^{i_n}
- r
- F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
+ \operatorname{ad}(F_{\beta_r})^{i_r}
+ u
+ F_{\beta_1}^{- i_1} \cdots F_{\beta_r}^{- i_r}
\]
- for all \(k_1, \ldots, k_n \in \mathbb{N}\).
+ for all \(k_1, \ldots, k_r \in \mathbb{N}\).
- Since the binomial coefficients \(\binom{x}{k} = \frac{x (x -1) \cdots (x - k +
- 1)}{k!}\) can be uniquely extended to polynomial functions in \(x \in K\), we
- may in general define
+ Since the binomial coefficients \(\binom{x}{k} = \frac{x (x-1) \cdots (x - k
+ + 1)}{k!}\) can be uniquely extended to polynomial functions in \(x \in K\),
+ we may in general define
\[
- \theta_\lambda(r)
- = \sum_{i_1, \ldots, i_n \ge 0}
- \binom{\lambda_1}{i_1} \cdots \binom{\lambda_n}{i_n}
+ \theta_\lambda(u)
+ = \sum_{i_1, \ldots, i_r \ge 0}
+ \binom{\lambda_1}{i_1} \cdots \binom{\lambda_r}{i_r}
\operatorname{ad}(F_{\beta_1})^{i_1} \cdots
- \operatorname{ad}(F_{\beta_n})^{i_n}
+ \operatorname{ad}(F_{\beta_r})^{i_r}
r
- F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
+ F_{\beta_1}^{- i_1} \cdots F_{\beta_r}^{- i_r}
\]
- for \(\lambda_1, \ldots, \lambda_n \in K\), \(\lambda = \lambda_1 \beta_1 +
- \cdots + \lambda_n \beta_n \in \mathfrak{h}^*\)
+ for \(\lambda_1, \ldots, \lambda_r \in K\), \(\lambda = \lambda_1 \beta_1 +
+ \cdots + \lambda_r \beta_r \in \mathfrak{h}^*\)
It is clear that the \(\theta_\lambda\) are endomorphisms. To see that the
\(\theta_\lambda\) are indeed automorphisms, notice \(\theta_{- k_1 \beta_1 -
- \cdots - k_n \beta_n} = \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}^{-1}\).
+ \cdots - k_r \beta_r} = \theta_{k_1 \beta_1 + \cdots + k_r \beta_r}^{-1}\).
The uniqueness of the polynomial extensions then implies \(\theta_{- \lambda}
- = \theta_\lambda^{-1}\) in general: given \(r \in \Sigma^{-1}
+ = \theta_\lambda^{-1}\) in general: given \(u \in \Sigma^{-1}
\mathcal{U}(\mathfrak{g})\), the map
\begin{align*}
\mathfrak{h}^* & \to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
- \lambda & \mapsto \theta_\lambda(\theta_{-\lambda}(r)) - r
+ \lambda & \mapsto \theta_\lambda(\theta_{-\lambda}(u)) - u
\end{align*}
- is a polynomial extension of the zero map \(\mathbb{Z} \beta_1 \oplus \cdots \oplus
- \mathbb{Z} \beta_n \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is therefore
- identically zero.
+ is a polynomial extension of the zero map \(\mathbb{Z} \beta_1 \oplus \cdots
+ \oplus \mathbb{Z} \beta_r \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is
+ therefore identically zero.
- Finally, let \(M\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
- whose restriction is a weight module. If \(m \in M\) then
+ Finally, let \(N\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
+ whose restriction is a weight module. If \(n \in N\) then
\[
- m \in (\theta_\lambda M)_{\mu + \lambda}
- \iff \theta_\lambda(H) m = (\mu + \lambda)(H) \cdot m
+ n \in (\theta_\lambda N)_{\mu + \lambda}
+ \iff \theta_\lambda(H) \cdot n = (\mu + \lambda)(H) n
\, \forall H \in \mathfrak{h}
\]
@@ -1287,44 +1286,45 @@ Explicitly\dots
and hence
\[
\begin{split}
- m \in (\theta_\lambda M)_{\mu + \lambda}
- & \iff (\lambda(H) + H) m = (\mu + \lambda)(H) \cdot m
+ n \in (\theta_\lambda N)_{\mu + \lambda}
+ & \iff (\lambda(H) + H) \cdot n = (\mu + \lambda)(H) n
\; \forall H \in \mathfrak{h} \\
- & \iff H m = \mu(H) \cdot m \; \forall H \in \mathfrak{h} \\
- & \iff m \in M_\mu
+ & \iff H \cdot n = \mu(H) n \; \forall H \in \mathfrak{h} \\
+ & \iff n \in N_\mu
\end{split},
\]
- so that \((\theta_\lambda M)_{\mu + \lambda} = M_\mu\).
+ so that \((\theta_\lambda N)_{\mu + \lambda} = N_\mu\).
\end{proof}
It should now be obvious\dots
\begin{proposition}[Mathieu]
- There exists a coherent extension \(\mathcal{M}\) of \(V\).
+ There exists a coherent extension \(\mathcal{M}\) of \(M\).
\end{proposition}
\begin{proof}
- Take\footnote{Here we fix some $\lambda_t \in t$ for each $Q$-coset $t \in
- \mfrac{\mathfrak{h}^*}{Q}$. While there is a natural isomorphism
- $\theta_\lambda \Sigma^{-1} V \isoto \theta_\mu \Sigma^{-1} V$ for each $\mu
- \in \lambda + Q$, they are not the same representation strictly speaking.
- This is yet another obstruction to the functoriality of our constructions.}
+ Take\footnote{Here we fix some $\lambda_\xi \in \xi$ for each $Q$-coset $\xi
+ \in \mfrac{\mathfrak{h}^*}{Q}$. While there is a natural isomorphism
+ $\theta_\lambda \Sigma^{-1} M \isoto \theta_\mu \Sigma^{-1} M$ for each $\mu
+ \in \lambda + Q$, they are not the same \(\mathfrak{g}\)-modules strictly
+ speaking. This is yet another obstruction to the functoriality of our
+ constructions.}
\[
\mathcal{M}
= \bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}
- \theta_\lambda \Sigma^{-1} V
+ \theta_\lambda \Sigma^{-1} M
\]
- It is clear \(V\) lies in \(\Sigma^{-1} V = \theta_0 \Sigma^{-1} V\) and
- therefore \(V \subset \mathcal{M}\). On the other hand, \(\dim
- \mathcal{M}_\mu = \dim \theta_\lambda \Sigma^{-1} V_\mu = \dim \Sigma^{-1}
- V_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\) -- \(\lambda\)
+ It is clear \(M\) lies in \(\Sigma^{-1} M = \theta_0 \Sigma^{-1} M\) and
+ therefore \(M \subset \mathcal{M}\). On the other hand, \(\dim
+ \mathcal{M}_\mu = \dim \theta_\lambda \Sigma^{-1} M_\mu = \dim \Sigma^{-1}
+ M_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\) -- \(\lambda\)
standing for some fixed representative of its \(Q\)-coset. Furthermore, given
\(u \in \mathcal{U}(\mathfrak{g})_0\) and \(\mu \in \lambda + Q\),
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
= \operatorname{Tr}
- (\theta_\lambda(u)\!\restriction_{\Sigma^{-1} V_{\mu - \lambda}})
+ (\theta_\lambda(u)\!\restriction_{\Sigma^{-1} M_{\mu - \lambda}})
\]
is polynomial in \(\mu\) because of the second item of
Proposition~\ref{thm:nice-automorphisms-exist}.
@@ -1334,30 +1334,30 @@ Lo and behold\dots
\begin{theorem}[Mathieu]
There exists a unique completely reducible coherent extension
- \(\mathcal{Ext}(V)\) of \(V\). More precisely, if \(\mathcal{M}\) is any
- coherent extension of \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong
- \mathcal{Ext}(V)\). Furthermore, \(\mathcal{Ext}(V)\) is
- a simple coherent family.
+ \(\mathcal{Ext}(M)\) of \(M\). More precisely, if \(\mathcal{M}\) is any
+ coherent extension of \(M\), then \(\mathcal{M}^{\operatorname{ss}} \cong
+ \mathcal{Ext}(M)\). Furthermore, \(\mathcal{Ext}(M)\) is
+ a irreducible coherent family.
\end{theorem}
\begin{proof}
The existence part should be clear from the previous discussion: it suffices
- to fix some coherent extension \(\mathcal{M}\) of \(V\) and take
- \(\mathcal{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
-
- To see that \(\mathcal{Ext}(V)\) is simple, recall from
- Corollary~\ref{thm:admissible-is-submod-of-extension} that \(V\) is a
- subrepresentation of \(\mathcal{Ext}(V)\). Since the degree of \(V\) is
- the same as the degree of \(\mathcal{Ext}(V)\), some of its weight
- spaces have maximal dimension inside of \(\mathcal{Ext}(V)\). In
- particular, it follows from Proposition~\ref{thm:centralizer-multiplicity}
- that \(\mathcal{Ext}(V)_\lambda = V_\lambda\) is a simple
+ to fix some coherent extension \(\mathcal{M}\) of \(M\) and take
+ \(\mathcal{Ext}(M) = \mathcal{M}^{\operatorname{ss}}\).
+
+ To see that \(\mathcal{Ext}(M)\) is irreducible, recall from
+ Corollary~\ref{thm:admissible-is-submod-of-extension} that \(M\) is a
+ \(\mathfrak{g}\)-submodule of \(\mathcal{Ext}(M)\). Since the degree of \(M\)
+ is the same as the degree of \(\mathcal{Ext}(M)\), some of its weight spaces
+ have maximal dimension inside of \(\mathcal{Ext}(M)\). In particular, it
+ follows from Proposition~\ref{thm:centralizer-multiplicity} that
+ \(\mathcal{Ext}(M)_\lambda = M_\lambda\) is a simple
\(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
- \operatorname{supp} V\).
+ \operatorname{supp} M\).
- As for the uniqueness of \(\mathcal{Ext}(V)\), fix some other completely
- reducible coherent extension \(\mathcal{N}\) of \(V\). We claim that the
- multiplicity of a given irreducible \(\mathfrak{g}\)-module \(W\) in
+ As for the uniqueness of \(\mathcal{Ext}(M)\), 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}
\begin{align*}
\mathfrak{h}^* \times \mathcal{U}(\mathfrak{g})_0 &
@@ -1366,35 +1366,35 @@ Lo and behold\dots
\mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})
\end{align*}
- It is a well known fact of the theory of modules that, given a \(K\)-algebra
- \(A\), a finite-dimensional completely reducible \(A\)-module \(M\) is
- determined, up to isomorphism, by its \emph{character}
+ 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}
\begin{align*}
- \chi_M : A & \to K \\
- a & \mapsto \operatorname{Tr}(a\!\restriction_M)
+ \chi_L : A & \to K \\
+ a & \mapsto \operatorname{Tr}(a\!\restriction_L)
\end{align*}
- In particular, the multiplicity of \(W\) in \(\mathcal{N}\), which is the
- same as the multiplicity of \(W_\lambda\) in \(\mathcal{N}_\lambda\), is
+ In particular, the multiplicity of \(L\) in \(\mathcal{N}\), which is the
+ same as the multiplicity of \(L_\lambda\) in \(\mathcal{N}_\lambda\), is
determined by the character \(\chi_{\mathcal{N}_\lambda} :
- \mathcal{U}(\mathfrak{g})_0 \to K\). Since this holds for all irreducible
+ \mathcal{U}(\mathfrak{g})_0 \to K\). Since this holds for all simple
weight \(\mathfrak{g}\)-modules, it follows that \(\mathcal{N}\) is
determined by its trace function. Of course, the same holds for
- \(\mathcal{Ext}(V)\). We now claim that the trace function of
- \(\mathcal{N}\) is the same as that of \(\mathcal{Ext}(V)\). Clearly,
- \(\operatorname{Tr}(u\!\restriction_{\mathcal{Ext}(V)_\lambda}) =
- \operatorname{Tr}(u\!\restriction_{V_\lambda}) =
+ \(\mathcal{Ext}(M)\). We now claim that the trace function of
+ \(\mathcal{N}\) is the same as that of \(\mathcal{Ext}(M)\). Clearly,
+ \(\operatorname{Tr}(u\!\restriction_{\mathcal{Ext}(M)_\lambda}) =
+ \operatorname{Tr}(u\!\restriction_{M_\lambda}) =
\operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\) for all \(\lambda
- \in \operatorname{supp}_{\operatorname{ess}} V\), \(u \in
- \mathcal{U}(\mathfrak{g})_0\). Since the essential support of \(V\) is
+ \in \operatorname{supp}_{\operatorname{ess}} M\), \(u \in
+ \mathcal{U}(\mathfrak{g})_0\). Since the essential support of \(M\) is
Zariski-dense and the maps \(\lambda \mapsto
- \operatorname{Tr}(u\!\restriction_{\mathcal{Ext}(V)_\lambda})\) and
+ \operatorname{Tr}(u\!\restriction_{\mathcal{Ext}(M)_\lambda})\) and
\(\lambda \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\)
are polynomial in \(\lambda \in \mathfrak{h}^*\), it follows that these maps
coincide for all \(u\).
- In conclusion, \(\mathcal{N} \cong \mathcal{Ext}(V)\) and
- \(\mathcal{Ext}(V)\) is unique.
+ In conclusion, \(\mathcal{N} \cong \mathcal{Ext}(M)\) and
+ \(\mathcal{Ext}(M)\) is unique.
\end{proof}
% This is a very important theorem, but since we won't classify the coherent
@@ -1403,12 +1403,12 @@ Lo and behold\dots
% are, which is a bit of a pain
%\begin{proposition}[Mathieu]
% The central characters of the irreducible submodules of
-% \(\operatorname{Ext}(V)\) are all the same.
+% \(\operatorname{Ext}(M)\) are all the same.
%\end{proposition}
-We have thus concluded our classification of cuspidal representations in terms
+We have thus concluded our classification of cuspidal modules 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
+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
@@ -1422,35 +1422,34 @@ results.
\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 \(\mathfrak{s}_1, \ldots, \mathfrak{s}_n\) are its simple component, then
- any irreducible weight \(\mathfrak{g}\)-module \(V\) can be decomposed as
+ \mathfrak{s}_r\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
+ and \(\mathfrak{s}_1, \ldots, \mathfrak{s}_r\) are its simple components,
+ then any simple weight \(\mathfrak{g}\)-module \(M\) can be decomposed as
\[
- V \cong Z \boxtimes V_1 \boxtimes \cdots \boxtimes V_n
+ M \cong Z \boxtimes M_1 \boxtimes \cdots \boxtimes M_r
\]
- where \(Z\) is a \(1\)-dimensional representation of \(\mathfrak{z}\) and \(V_i\)
- is an irreducible weight \(\mathfrak{s}_i\)-module.
+ where \(Z\) is a \(1\)-dimensional representation of \(\mathfrak{z}\) and
+ \(M_i\) 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 \(\mathfrak{s} \cong
- \mathfrak{sp}_{2 n}(K)\).
+ there exists a simple cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s}
+ \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\).
\end{proposition}
We have previously seen that the representations of Abelian Lie algebras,
-particularly the \(1\)-dimensional ones, are well understood. Hence 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
+particularly the \(1\)-dimensional ones, are well understood. Hence to classify
+the simple modules of an arbitrary reductive algebra it suffices to classify
+those of its simple components. To classify these module we can apply
+Fernando's results and reduce the problem to constructing the cuspidal modules
+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
+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 simple completely reducible coherent families of
+classifying the irreducible completely reducible 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