- Commit
- 4486d7c077ade408dc6a03ef096d88f0604397c3
- Parent
- 3e2e450894d132e16c1b40e0acacf12315cf9c14
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Revised the discussion on Fernando's work
Elaborated on the motivation behind parabolic induction and fixed some mistakes contained in the text
Also removed unnecessary comments on notation
Also tweaked the definition of cuspidal modules: a cuspidal module is now a simple weight module by definition
Also added a TODO item
diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex
@@ -47,7 +47,7 @@ elements outside of \(\mathfrak{h}\) left to analyze.
Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
\subset \mathfrak{g}\), which leads us to the following definition.
-\begin{definition}\index{Cartan subalgebra}
+\begin{definition}\index{Lie subalgebra!Cartan subalgebra}
A subalgebra \(\mathfrak{h} \subset \mathfrak{g}\) is called \emph{a Cartan
subalgebra of \(\mathfrak{g}\)} if is self-normalizing -- i.e. \([X, H] \in
\mathfrak{h}\) for all \(H \in \mathfrak{h}\) if, and only if \(X \in
@@ -512,13 +512,15 @@ induces an order in \(Q\), where elements are ordered by their
respectively.
\end{definition}
-\begin{definition}
+\begin{definition}\index{Lie subalgebra!Borel subalgebra}\index{Lie subalgebra!parabolic subalgebra}
Let \(\Sigma\) be a basis for \(\Delta\). The subalgebra \(\mathfrak{b} =
\mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta^+} \mathfrak{g}_\alpha\) is
called \emph{the Borel subalgebra associated with \(\mathfrak{h}\) and
- \(\Sigma\)}.
+ \(\Sigma\)}. A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called
+ \emph{parabolic} if \(\mathfrak{p} \supset \mathfrak{b}\).
\end{definition}
+% TODO: This is a total order on Q, not a partial order
It should be obvious that the binary relation \(\preceq\) in \(Q\) is a partial
order. In addition, we may compare the elements of a given \(Q\)-coset
\(\lambda + Q\) by comparing their difference with \(0 \in Q\). In other words,
@@ -680,11 +682,12 @@ this condition is also sufficient. In other words\dots
\begin{definition}\index{weights!dominant weight}\index{weights!integral weight}
An element \(\lambda\) of \(P\) such that \(\lambda(H_\alpha) \ge 0\) for all
\(\alpha \in \Delta^+\) is referred to as an \emph{dominant integral weight
- of \(\mathfrak{g}\)}.
+ of \(\mathfrak{g}\)}. The set of all dominant integral weights is denotes by
+ \(P^+\).
\end{definition}
\begin{theorem}\label{thm:dominant-weight-theo}\index{weights!Highest Weight Theorem}
- For each dominant integral \(\lambda \in P\) there exists precisely one
+ For each dominant integral \(\lambda \in P^+\) there exists precisely one
finite-dimensional simple \(\mathfrak{g}\)-module \(M\) whose highest weight
is \(\lambda\).
\end{theorem}
@@ -1035,7 +1038,7 @@ Indeed\dots
All its left to prove the Highest Weight Theorem is verifying that the
situation encountered in Example~\ref{ex:sl2-verma-quotient} holds for any
-dominant \(\lambda \in P\). In other words, we need to show\dots
+dominant integral \(\lambda \in P^+\). In other words, we need to show\dots
\begin{proposition}\label{thm:verma-is-finite-dim}
If \(\mathfrak{g}\) is semisimple and \(\lambda\) is dominant integral then
@@ -1071,9 +1074,9 @@ ready to prove the Highest Weight Theorem.
\end{proof}
We would now like to conclude this chapter by describing the situation where
-\(\lambda \in \mathfrak{h}^*\) is not dominant integral. We begin by pointing
-out that Proposition~\ref{thm:verma-is-finite-dim} fails in the general
-setting. For instance, consider\dots
+\(\lambda \notin P^+\). We begin by pointing out that
+Proposition~\ref{thm:verma-is-finite-dim} fails in the general setting. For
+instance, consider\dots
\begin{example}\label{ex:antidominant-verma}
The action of \(\mathfrak{sl}_2(K)\) on \(M(-4)\) is given by the following
@@ -1093,13 +1096,12 @@ setting. For instance, consider\dots
\end{example}
While \(L(\lambda)\) is always a highest weight module of highest weight
-\(\lambda\), one can show that if \(\lambda\) is not dominant integral then
-\(L(\lambda)\) is infinite-dimensional. Indeed, since the highest weight of a
-finite-dimensional simple \(\mathfrak{g}\)-module is always dominant integral,
-\(L(\lambda)\) is infinite-dimensional for any \(\lambda\) which is not
-dominant integral. If \(\lambda = k_1 \beta_1 + \cdots + k_r \beta_r \in P\) is
-integral and \(k_i < 0\) for all \(i\), then \(M(\lambda) \cong L(\lambda)\) as
-in Example~\ref{ex:antidominant-verma}.
+\(\lambda\), one can show that if \(\lambda \notin P^+\) then \(L(\lambda)\) is
+infinite-dimensional. Indeed, since the highest weight of a finite-dimensional
+simple \(\mathfrak{g}\)-module is always dominant integral, \(L(\lambda)\) is
+infinite-dimensional for any \(\lambda \notin P^+\). If \(\lambda = k_1 \beta_1
++ \cdots + k_r \beta_r \in P\) is integral and \(k_i < 0\) for all \(i\), then
+\(M(\lambda) \cong L(\lambda)\) as in Example~\ref{ex:antidominant-verma}.
Verma modules can thus serve as examples of infinite-dimensional simple
modules. In the next chapter we expand our previous results by exploring the
diff --git a/sections/simple-weight.tex b/sections/simple-weight.tex
@@ -3,7 +3,7 @@
In this chapter we will expand our results on finite-dimensional simple modules
of semisimple Lie algebras by considering \emph{infinite-dimensional}
\(\mathfrak{g}\)-modules, which introduces numerous complications to our
-analysis.
+analysis.
For instance, in the infinite-dimensional setting we can no longer take
complete-reducibility for granted. Indeed, we have seen that even if
@@ -182,7 +182,7 @@ A particularly well behaved class of examples are the so called
\(\mathfrak{g}_i\)-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
+ \cdot \deg M_2\) and
\[
\operatorname{supp}_{\operatorname{ess}} M_1 \otimes M_2
= \operatorname{supp}_{\operatorname{ess}} M_1 \oplus
@@ -258,124 +258,129 @@ 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.
-
-% 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.
-
-% TODO: Move this definition to just after the definition of the Borel
-% subalgebra
-\begin{definition}\index{Lie subalgebra!parabolic subalgebra}
- A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
- 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
-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}(M) \\
- \mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})} \arrow[dotted]{ur} &
- \end{tikzcd}
-\end{center}
-
-Let \(\mathfrak{p}\) be a parabolic subalgebra and \(M\) be a simple
-weight \(\mathfrak{p}\)-module. We should point out that while
-\(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} M\) 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.
+solution to our problem, we consider the Verma modules \(M(\lambda)\) for
+\(\lambda \in \mathfrak{h}^*\) which is not dominant integral. After all, the
+simple quotients of Verma modules form a remarkably large class of
+infinite-dimensional simple weight modules -- at least as large as
+\(\mathfrak{h}^* \setminus P^+\)! More generally, the induction functor
+\(\operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} :
+\mathfrak{b}\text{-}\mathbf{Mod} \to \mathfrak{g}\text{-}\mathbf{Mod}\) has
+proven itself a powerful tool for constructing modules.
+
+We claim this is not an unmotivated guess. Specifically, there are very good
+reasons behind the choice to consider induction over the Borel subalgebra
+\(\mathfrak{b} \subset \mathfrak{g}\). First, the fact that \(\mathfrak{h}
+\subset \mathfrak{g}\) affords us great control over the weight spaces of
+\(\operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} M\): by assigning a
+prescribed action of \(\mathfrak{h}\) to \(M\) we can ensure that
+\(\operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} M = \bigoplus_\lambda
+(\operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} M)_\lambda\). In addition, we
+have seen in the proof of Proposition~\ref{thm:high-weight-mod-is-weight-mod}
+that by requiring that the positive part of \(\mathfrak{b}\) acts on \(M\) by
+zero we can ensure that \(\dim
+(\operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} M)_\lambda < \infty\). All in
+all, the nature of \(\mathfrak{b}\) affords us just enough control to guarantee
+that \(\operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} M\) is a weight module
+for sufficiently well behaved \(M\).
+
+Unfortunately for us, this is still too little control: there are simple weight
+modules which are not of the form \(L(\lambda)\). More generally, we may
+consider induction over some parabolic subalgebra \(\mathfrak{p} \subset
+\mathfrak{g}\) -- i.e. some subalgebra such that \(\mathfrak{p} \supset
+\mathfrak{g}\). This leads us to the following definition.
\begin{definition}\index{\(\mathfrak{g}\)-module!(generalized) Verma modules}
- 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}.
+ Let \(\mathfrak{p} \subset \mathfrak{g}\) be a parabolic subalgebra and \(M\)
+ be a simple \(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module. We
+ can view \(M\) as a \(\mathfrak{p}\)-module where
+ \(\mathfrak{nil}(\mathfrak{p})\) acts by zero by setting \(X \cdot m = (X +
+ \mathfrak{nil}(\mathfrak{p})) \cdot m\) for all \(m \in M\) and \(X \in
+ \mathfrak{p}\) -- which is the same as the \(\mathfrak{p}\)-module given by
+ composing the action map \(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}
+ \to \mathfrak{gl}(M)\) with the projection \(\mathfrak{p} \to
+ \mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\). The module
+ \(M_{\mathfrak{p}}(M) = \operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} M\)
+ is called \emph{generalized Verma module associated with \(M\)}.
\end{definition}
-\begin{proposition}\label{thm:generalized-verma-has-simple-quotient}
- 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}
+\begin{example}
+ It is not hard to see that
+ \(\mfrac{\mathfrak{b}}{\mathfrak{nil}(\mathfrak{b})} = \mathfrak{h}\). If we
+ take \(\lambda \in \mathfrak{h}^*\) and let \(K m^+\) be the
+ \(1\)-dimensional \(\mathfrak{h}\)-module where \(\mathfrak{h}\) acts by
+ \(\lambda\) then \(M(\lambda) = M_{\mathfrak{b}}(K m^+)\).
+\end{example}
-The proof of Proposition~\ref{thm:generalized-verma-has-simple-quotient} is
-entirely analogous to that of Proposition~\ref{thm:max-verma-submod-is-weight}.
-This leads us to the following definitions.
+As promised, \(M_{\mathfrak{p}}(M)\) is generally well behaved for well behaved
+\(M\). In particular, if \(M\) is highest weight
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module then
+\(M_{\mathfrak{p}}(M)\) is also a highest weight \(\mathfrak{g}\)-module, and
+if \(M\) is a weight
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module then
+\(M_{\mathfrak{p}}(M)\) is a weight module with \(M_{\mathfrak{p}}(M)_\lambda =
+\sum_{\alpha + \mu = \lambda} \mathcal{U}(\mathfrak{g})_\alpha
+\otimes_{\mathcal{U}(\mathfrak{p})} M_\mu\), -- see Lemma 1.1 of \cite{mathieu}
+for a full proof. However, \(M_{\mathfrak{p}}(M)\) is not simple in general.
+Indeed, regular Verma modules not necessarily simple. This issue may be dealt
+with by passing to the simple quotients of \(M_{\mathfrak{p}}(M)\).
+
+Let \(M\) be a simple weight
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module. As it turns out,
+the situation encountered in Proposition~\ref{thm:max-verma-submod-is-weight}
+is also verified in the general setting. Namely, since \(M_{\mathfrak{p}}(M)\)
+is generated by \(K \otimes_{\mathcal{U}(\mathfrak{p})} M = \bigoplus_{\lambda
+\in Q + \operatorname{supp} M} M_{\mathfrak{p}}(M)_\lambda\), it follows that
+any proper submodule of \(M_{\mathfrak{p}}(M)\) is contained in
+\(\bigoplus_{\lambda \notin Q + \operatorname{supp} M}
+M_{\mathfrak{p}}(M)_\lambda\). The sum \(N_{\mathfrak{p}}(M)\) of all such
+submodules is thus the unique maximal submodule of \(M_{\mathfrak{p}}(M)\) and
+\(L_{\mathfrak{p}}(M) = \mfrac{M_{\mathfrak{p}}(M)}{N_{\mathfrak{p}}(M)}\) is
+its unique simple quotient -- again, we refer the reader to \cite{mathieu} for
+a complete proof. This leads us to the following definition.
\begin{definition}\index{\(\mathfrak{g}\)-module!parabolic induced modules}\index{\(\mathfrak{g}\)-module!cuspidal modules}
- 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
+ A simple weight \(\mathfrak{g}\)-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 simple weight
+ \(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module \(M\). A
+ \emph{cuspidal \(\mathfrak{g}\)-module} is a simple weight
\(\mathfrak{g}\)-module which is \emph{not} parabolic induced.
\end{definition}
-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
+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 by a cuspidal module.
\begin{theorem}[Fernando]
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 \(M\).
+ \mathfrak{g}\) and some cuspidal
+ \(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module \(M\).
\end{theorem}
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
+\(\mathfrak{g}\)-modules and pairs \((\mathfrak{p}, M)\) 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, \(M\) 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 \(\mathfrak{p}' =
- \twisted{\mathfrak{p}}{\sigma}\) and \(M \cong \twisted{N}{\sigma}\) for
- some\footnote{Here $\twisted{\mathfrak{p}}{\sigma}$ denotes the image of
- $\mathfrak{p}$ under the automorphism of $\sigma : \mathfrak{g} \to
- \mathfrak{g}$ given by the canonical action of $W$ on $\mathfrak{g}$ and
- $\twisted{N}{\sigma}$ is the $\mathfrak{p}$-module given by composing the map
- $\mathfrak{p}' \to \mathfrak{gl}(N)$ with the restriction
- $\sigma\!\restriction_{\mathfrak{p}} : \mathfrak{p} \to \mathfrak{p}'$.}
- \(\sigma \in W_M\), where
+ \subset \mathfrak{g}\) is another parabolic subalgebra, \(M\) is a cuspidal
+ \(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module and \(N\) is a
+ cuspidal \(\mfrac{\mathfrak{p}'}{\mathfrak{nil}(\mathfrak{p}')}\)-module then
+ \(L_{\mathfrak{p}}(M) \cong L_{\mathfrak{p}'}(N)\) if, and only if
+ \(\mathfrak{p}' = \twisted{\mathfrak{p}}{\sigma}\) and \(M \cong
+ \twisted{N}{\sigma}\) as \(\mathfrak{p}\)-modules for some\footnote{Here
+ $\twisted{\mathfrak{p}}{\sigma}$ denotes the image of $\mathfrak{p}$ under
+ the automorphism of $\sigma : \mathfrak{g} \to \mathfrak{g}$ given by the
+ canonical action of $W$ on $\mathfrak{g}$ and $\twisted{N}{\sigma}$ is the
+ $\mathfrak{p}$-module given by composing the map $\mathfrak{p}' \to
+ \mathfrak{gl}(N)$ with the restriction $\sigma\!\restriction_{\mathfrak{p}} :
+ \mathfrak{p} \to \mathfrak{p}'$.} \(\sigma \in W_M\), where
\[
W_M
= \langle
@@ -401,11 +406,8 @@ characterizations of cuspidal modules.
\begin{enumerate}
\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} 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.
+ \(\alpha \in \Delta\).
+ \item The support of \(M\) is precisely one \(Q\)-coset.
\end{enumerate}
\end{corollary}
@@ -415,12 +417,12 @@ characterizations of cuspidal modules.
Hence \(K[x, x^{-1}]\) is a cuspidal \(\mathfrak{sl}_2(K)\)-module.
\end{example}
-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
+Having reduced our classification problem to that of classifying 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}
@@ -506,7 +508,7 @@ act injectively in \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\), so that
\(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) is simple. In particular, if
\(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with \(\lambda \notin \mu + 2
\mathbb{Z}\) then \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) and
-\(\twisted{K[x, x^{-1}]}{\varphi_\mu}\) are non-isomorphic simple cuspidal
+\(\twisted{K[x, x^{-1}]}{\varphi_\mu}\) are non-isomorphic cuspidal
\(\mathfrak{sl}_2(K)\)-modules, since their supports differ. These cuspidal
modules can be ``glued together'' in a \emph{monstrous concoction} by summing
over \(\lambda \in K\), as in
@@ -586,11 +588,11 @@ families}.
notion of coherent families and the so called \emph{weighting functor}.
\end{note}
-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}. In
-addition, we hope that such coherent families are somehow \emph{uniquely
-determined} by \(M\). This leads us to the following definition.
+Our hope is that given a 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}. 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
@@ -600,11 +602,11 @@ determined} by \(M\). This leads us to the following definition.
Our goal is now showing that every simple bounded 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 simple bounded \(\mathfrak{g}\)-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.
+submodules of a given coherent family are actually cuspidal modules. If every
+simple bounded \(\mathfrak{g}\)-module fits inside a coherent extension, this
+would lead to classification of all 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
@@ -874,7 +876,7 @@ computed via the following lemma.
\end{proof}
A complementary question now is: which submodules of a \emph{nice} coherent
-family are cuspidal representations?
+family are cuspidal?
\begin{proposition}[Mathieu]
Let \(\mathcal{M}\) be an irreducible coherent family of degree \(d\) and
@@ -1593,8 +1595,8 @@ Specifically\dots
\begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra. Suppose
- there exists a simple cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s}
- \cong \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\).
+ there exists a cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s} \cong
+ \mathfrak{sl}_n(K)\) or \(\mathfrak{s} \cong \mathfrak{sp}_{2 n}(K)\).
\end{proposition}
Hence it suffices to classify the irreducible semisimple coherent families of