- Commit
- eb986e432fbd9ae3d07e4d5cd89a4f48ec017c49
- Parent
- c389afd8338939bf5bd033d8d012b94ef4127489
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Added a proof of a proposition
Added a proof of the proposition which established the relationship between the multiplicities of simple weight modules in semisimple weight modules and the multiplicities of the correspoding weight spaces
This result is a slight generalization of the reult proved in Mathieu's paper, which is needed to state some comments on cohenret families of semisimple algebras we will add later on
diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -49,7 +49,7 @@ Since the weight space decomposition was perhaps the single most instrumental
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}\index{\(\mathfrak{g}\)-module!weight modules}\index{weights!weight modules}\index{\(\mathfrak{g}\)-module!(essential) support}
+\begin{definition}\label{def:weight-mod}\index{\(\mathfrak{g}\)-module!weight modules}\index{weights!weight modules}\index{\(\mathfrak{g}\)-module!(essential) support}
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
@@ -64,6 +64,16 @@ to the case it holds. This brings us to the following definition.
Lie algebra is a weight module.
\end{example}
+\begin{example}\label{ex:reductive-alg-equivalence}
+ We have seen that every finite-dimensional \(\mathfrak{g}\)-module is a
+ weight module for semisimple \(\mathfrak{g}\). In particular, if
+ \(\mathfrak{g}\) is finite-dimensional then the adjoint
+ \(\mathfrak{g}\)-module \(\mathfrak{g}\) is a weight module. More generally,
+ a finite-dimensional Lie algebra \(\mathfrak{g}\) is reductive if, and only if
+ the adjoint \(\mathfrak{g}\)-module \(\mathfrak{g}\) is a weight module, in
+ which case its weight spaces are given by the root spaces of \(\mathfrak{g}\)
+\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
@@ -119,6 +129,128 @@ to the case it holds. This brings us to the following definition.
\(M_i\) is a simple weight \(\mathfrak{s}_i\)-module.
\end{example}
+\begin{example}\label{ex:adjoint-action-in-universal-enveloping-is-weight}
+ We would like to show that the requirement of finite-dimensionality in
+ Definition~\ref{def:weight-mod} is not redundant. Let \(\mathfrak{g}\) be a
+ finite-dimensional reductive Lie algebra and consider the adjoint
+ \(\mathfrak{g}\)-module \(\mathcal{U}(\mathfrak{g})\) -- where \(X \in
+ \mathfrak{g}\) acts by taking commutators. Given \(\alpha \in Q\), a simple
+ computation shows \(K \langle X_1 \cdots X_n H_1 \cdots H_m : X_i \in
+ \mathfrak{g}_{\alpha_i}, H_i \in \mathfrak{h}, \alpha_i \in \Delta, \alpha =
+ \alpha_1 + \cdots + \alpha_n \rangle \subset
+ \mathcal{U}(\mathfrak{g})_\alpha\). The Poincaré-Birkoff-Witt Theorem and
+ Example~\ref{ex:reductive-alg-equivalence} thus imply that
+ \(\mathcal{U}(\mathfrak{g}) = \bigoplus_{\alpha \in Q}
+ \mathcal{U}(\mathfrak{g})_\alpha\) where \(\mathcal{U}(\mathfrak{g})_\alpha =
+ K \langle X_1 \cdots X_n H_1 \cdots H_m : X_i \in \mathfrak{g}_{\alpha_i},
+ H_i \in \mathfrak{h}, \alpha_i \in \Delta, \alpha = \alpha_1 + \cdots +
+ \alpha_n \rangle\). However, \(\dim \mathcal{U}(\mathfrak{g})_\alpha =
+ \infty\). For instance, \(\mathcal{U}(\mathfrak{g})_0\) is \emph{precisely}
+ the commutator of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\), which
+ contains \(\mathcal{U}(\mathfrak{h})\) and is therefore infinite-dimensional.
+\end{example}
+
+We would like to stress that the weight spaces \(M_\lambda \subset M\) are
+\emph{not} \(\mathfrak{g}\)-submodules. Nevertheless, \(M_\lambda\) is a
+\(\mathfrak{h}\)-submodule. More generally, \(M_\lambda\) is a
+\(\mathcal{U}(\mathfrak{g})_0\)-submodule, where
+\(\mathcal{U}(\mathfrak{g})_0\) is the centralizer of \(\mathfrak{h}\) in
+\(\mathcal{U}(\mathfrak{g})\), and the multiplicities of simple
+\(\mathfrak{g}\)-modules in a semisimple weight \(\mathfrak{g}\)-module \(M\)
+are related to the multiplicities of simple
+\(\mathcal{U}(\mathfrak{g})_0\)-modules in \(M_\lambda\) via the following
+result.
+
+\begin{proposition}\label{thm:centralizer-multiplicity}
+ Let \(\mathfrak{g}\) be a finite-dimensional reductive Lie algebra and \(M\)
+ be a semisimple weight \(\mathfrak{g}\)-module. Then \(M_\lambda\) is a
+ semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
+ \operatorname{supp} M\). Moreover, if \(L\) is a simple weight
+ \(\mathfrak{g}\)-module such that \(\lambda \in \operatorname{supp} L\) then
+ \(L_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module and the
+ multiplicity \(L\) in \(M\) coincides with the multiplicity of \(L_\lambda\)
+ in \(M_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module.
+\end{proposition}
+
+\begin{proof}
+ We begin by showing that \(L_\lambda\) is simple. Let \(N \subset L_\lambda\)
+ be a nontrivial \(\mathcal{U}(\mathfrak{g})_0\)-submodule. We want to
+ establish that \(N = L_\lambda\).
+
+ If \(\mathcal{U}(\mathfrak{g})_\alpha\) denotes the root space of \(\alpha\)
+ in \(\mathcal{U}(\mathfrak{g})\) under the adjoint action of \(\mathfrak{g}\)
+ as in Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight},
+ \(\alpha \in Q\), a simple calculation shows
+ \(\mathcal{U}(\mathfrak{g})_\alpha \cdot N \subset L_{\lambda + \alpha}\).
+ Since \(L\) is simple and \(N\) is nonzero, it follows from
+ Example~\ref{ex:adjoint-action-in-universal-enveloping-is-weight} that
+ \[
+ L
+ = \mathcal{U}(\mathfrak{g}) \cdot N
+ = \bigoplus_{\alpha \in Q} \mathcal{U}(\mathfrak{g})_\alpha \cdot N
+ \]
+ and thus \(L_{\lambda + \alpha} = \mathcal{U}(\mathfrak{g})_\alpha \cdot N\).
+ In particular, \(L_\lambda = \mathcal{U}(\mathfrak{g})_0 \cdot N \subset N\)
+ and \(N = L_\lambda\).
+
+ Now given a semisimple weight \(\mathfrak{g}\)-module \(M = \bigoplus_i M_i\)
+ with \(M_i\) simple, it is clear \(M_\lambda = \bigoplus_i (M_i)_\lambda\).
+ Each \((M_i)_\lambda\) is either \(0\) or a simple
+ \(\mathcal{U}(\mathfrak{g})_0\)-module, so that \(M_\lambda\) is a semisimple
+ \(\mathcal{U}(\mathfrak{g})_0\)-module. In addition, to see that the
+ multiplicity of \(L\) in \(M\) coincides with the multiplicity of
+ \(L_\lambda\) in \(M_\lambda\) it suffices to show that if \((M_i)_\lambda
+ \cong (M_j)_\lambda\) are both nonzero then \(M_i \cong M_j\).
+
+ If \(I(M_i) = \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{g})_0}
+ (M_i)_\lambda\), the inclusion of \(\mathcal{U}(\mathfrak{g})_0\)-modules
+ \((M_i)_\lambda \to M_i\) induces a \(\mathfrak{g}\)-homomorphism
+ \begin{align*}
+ I(M_i) & \to M_i \\
+ u \otimes m & \mapsto u \cdot m
+ \end{align*}
+
+ Since \(M_i\) is simple and \(\lambda \in \operatorname{supp} M_i\), \(M_i =
+ \mathcal{U}(\mathfrak{g}) \cdot (M_i)_\lambda\). The homomorphism \(I(M_i)
+ \to M_i\) is thus surjective. Similarly, if \(I(M_j) =
+ \mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{g})_0}
+ (M_j)_\lambda\) then there is a natural surjective
+ \(\mathfrak{g}\)-homomorphism \(I(M_j) \to M_j\). Now suppose there is an
+ isomorphism of \(\mathcal{U}(\mathfrak{g})_0\)-modules \(f: (M_i)_\lambda
+ \isoto (M_j)_\lambda\). Such an isomorphism induces an isomorphism of
+ \(\mathfrak{g}\)-modules
+ \begin{align*}
+ \tilde f : I(M_i) & \isoto I(M_j) \\
+ u \otimes m & \mapsto u \otimes f(m)
+ \end{align*}
+
+ By composing \(\tilde f\) with the projection \(I(M_j) \to M_j\) we get a
+ surjective homomorphism \(I(M_i) \to M_j\). We claim \(\ker (I(M_i) \to M_i)
+ = \ker (I(M_i) \to M_j)\). To see this, notice that \(\ker(I(M_i) \to M_i)\)
+ coincides with the largest submodule \(Z(M_i) \subset I(M_i)\) contained in
+ \(\bigoplus_{\alpha \ne 0} \mathcal{U}(\mathfrak{g})_\alpha
+ \otimes_{\mathcal{U}(\mathfrak{g})_0} (M_i)_\lambda\). Indeed, a simple
+ computation shows \(\ker (I(M_i) \to M_i) \cap (\mathcal{U}(\mathfrak{g})_0
+ \otimes_{\mathcal{U}(\mathfrak{g})_0} (M_i)_\lambda) = 0\), which implies
+ \(\ker(I(M_i) \to M_i) \subset Z(M_i)\). Since \(M_i\) is simple, \(\ker
+ (I(M_i) \to M_i)\) is maximal and thus \(\ker(I(M_i) \to M_i) = Z(M_i)\). By
+ the same token, \(\ker (I(M_j) \to M_j)\) is the largest submodule of
+ \(I(M_j)\) contained in \(\bigoplus_{\alpha \ne 0}
+ \mathcal{U}(\mathfrak{g})_\alpha \otimes_{\mathcal{U}(\mathfrak{g})_0}
+ (M_j)_\lambda\) and therefore \(\ker(I(M_i) \to M_i) =
+ \tilde{f}^{-1}(\ker(I(M_j) \to M_j)) = \ker(I(M_i) \to M_j)\).
+
+ Hence there is an isomorphism \(\mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \isoto
+ M_j\) satisfying
+ \begin{center}
+ \begin{tikzcd}
+ I(M_i) \rar{\tilde f} \dar & I(M_j) \dar \\
+ \mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \rar{\sim} & M_j
+ \end{tikzcd}
+ \end{center}
+ and finally \(M_i \cong \mfrac{I(M_i)}{\ker(I(M_i) \to M_i)} \cong M_j\).
+\end{proof}
+
A particularly well behaved class of examples are the so called
\emph{bounded} modules.
@@ -464,13 +596,7 @@ families}.
\item \(\dim \mathcal{M}_\lambda = d\) for \emph{all} \(\lambda \in
\mathfrak{h}^*\) -- i.e. \(\operatorname{supp}_{\operatorname{ess}}
\mathcal{M} = \mathfrak{h}^*\).
- \item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the
- centralizer\footnote{The notation $\mathcal{U}(\mathfrak{g})_0$ for the
- centralizer of $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ comes from
- the fact it coincides with the weight space associated with $0 \in
- \mathfrak{h}^*$ in the adjoint action of $\mathfrak{g}$ on
- $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the regular action
- of $\mathfrak{g}$ on $\mathcal{U}(\mathfrak{g})$.}
+ \item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the centralizer
\(\mathcal{U}(\mathfrak{g})_0\) of \(\mathfrak{h}\) in
\(\mathcal{U}(\mathfrak{g})\), the map
\begin{align*}
@@ -705,18 +831,7 @@ These last results provide a partial answer to the question of existence of
well behaved coherent extensions. A complementary question now is: which
submodules of a \emph{nice} coherent family are cuspidal representations?
-\begin{proposition}\label{thm:centralizer-multiplicity}
- Let \(M\) be a semisimple weight \(\mathfrak{g}\)-module. Then \(M_\lambda\)
- is a semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
- \mathfrak{h}^*\), 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]
+\begin{proposition}[Mathieu]
Let \(\mathcal{M}\) be an irreducible coherent family of degree \(d\) and
\(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent.
\begin{enumerate}
@@ -725,7 +840,7 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
all \(\alpha \in \Delta\).
\item \(\mathcal{M}[\lambda]\) is cuspidal.
\end{enumerate}
-\end{corollary}
+\end{proposition}
\begin{proof}
The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
@@ -762,9 +877,7 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
\(\mathcal{M}[\lambda] = M\) and \(\mathcal{M}[\lambda]\) is cuspidal.
\end{proof}
-Once more, the proof of Proposition~\ref{thm:centralizer-multiplicity} wasn't
-deemed informative enough to be included in here, but see the proof of Lemma
-2.3 of \cite{mathieu}. To finish the proof, we now show\dots
+To finish the proof, we now show\dots
\begin{lemma}
Let \(\mathcal{M}\) be a coherent family. The set \(U = \{\lambda \in