- Commit
- 51e9c3007ff31a835c1a6bfe39bdb8aac2251db1
- Parent
- e439b14d73a79d1c7006a8da078c5aa988d10213
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Finished hydrating the section on the basics of coherent families and coherent extensions
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Finished hydrating the section on the basics of coherent families and coherent extensions
1 file changed, 137 insertions, 95 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/mathieu.tex | 232 | 137 | 95 |
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -475,7 +475,7 @@ 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 engineous breaktrough was the realization that -\(\mathcal{M}\) is a particular example of a more general pattern, which ha +\(\mathcal{M}\) is a particular example of a more general pattern, which he named \emph{coherent families}. \begin{definition} @@ -517,7 +517,32 @@ named \emph{coherent families}. \(x^k\) we can see \((\mathcal{M}(\sfrac{1}{2}))[0] \cong K[x, x^{-1}]\). \end{example} -% TODO: Move this somewhere else +Our hope is that given an irreducible cuspidal representation \(V\), we can +somehow find \(V\) 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. + +\begin{definition} + Given an admissible representation \(V\) of \(\mathfrak{g}\) of degree \(d\), + a coherent extension \(\mathcal{M}\) of \(V\) is a coherent family + \(\mathcal{M}\) of degree \(d\) that contains \(V\) 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 out +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. Idealy, 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 +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 @@ -534,86 +559,86 @@ named \emph{coherent families}. \mathfrak{h}^*\). \end{definition} -\begin{definition} - Given an admissible representation \(V\) of \(\mathfrak{g}\) of degree \(d\), - a coherent extension \(\mathcal{M}\) of \(V\) is a coherent family - \(\mathcal{M}\) of degree \(d\) that contains \(V\) as a subquotient. -\end{definition} +Another natural cadidate for the role of ``nice extensions'' are the completely +reducible coherent families -- i.e. families wich 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 synonim 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 \(V\). +% TODO: Note somewhere that M[mu] is a submodule % Mathieu's proof of this is somewhat profane, I don't think it's worth % including it in here -% TODO: Define the notation for M[mu] somewhere else -% TODO: Note somewhere that M[mu] is a submodule \begin{lemma} Given a coherent family \(\mathcal{M}\) and \(\lambda \in \mathfrak{h}^*\), \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module. \end{lemma} -% TODO: From this we may conclude that any admissible submodule is a submodule -% of the semisimplification of any of its coherent extensions +% TODO: Point out this construction is NOT functorial, since it depends on the +% choice of composition series \begin{corollary} Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a unique completely reducible coherent family \(\mathcal{M}^{\operatorname{ss}}\) of degree \(d\) such that the composition series of \(\mathcal{M}^{\operatorname{ss}}[\lambda]\) is the same as that of \(\mathcal{M}[\lambda]\) for all \(\lambda \in \mathfrak{h}^*\), called - \emph{the semisimplification\footnote{Recall that a ``semisimple'' is a - synonim for ``completely reducible'' in the context of modules.} of - \(\mathcal{M}\)}. - - Namely, if \(\{\lambda_i\}_i\) is a set of representatives of the - \(Q\)-cosets of \(\mathfrak{h}^*\) and \(0 = \mathcal{M}_{i 0} \subset - \mathcal{M}_{i 1} \subset \cdots \subset \mathcal{M}_{i n_i} = - \mathcal{M}[\lambda_i]\) is a composition series, + \emph{the semisimplification of \(\mathcal{M}\)}. + + 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, \[ \mathcal{M}^{\operatorname{ss}} - \cong \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}} + \cong \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}} + \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}} \] \end{corollary} \begin{proof} The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear: since \(\mathcal{M}^{\operatorname{ss}}\) is completely reducible, so is - \(\mathcal{M}^{\operatorname{ss}}[\lambda_i]\). Hence + \(\mathcal{M}^{\operatorname{ss}}[\lambda]\). Hence \[ - \mathcal{M}^{\operatorname{ss}}[\lambda_i] - \cong \bigoplus_j \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}} + \mathcal{M}^{\operatorname{ss}}[\lambda] + \cong + \bigoplus_i \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}} \] As for the existence of the semisimplification, it suffices to show \[ \mathcal{M}^{\operatorname{ss}} - = \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}} + = \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}} + \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}} \] is indeed a completely reducible coherent family of degree \(d\). We know from examples~\ref{ex:submod-is-weight-mod} and - \ref{ex:quotient-is-weight-mod} that each quotient \(\mfrac{\mathcal{M}_{i j - + 1}}{\mathcal{M}_{i j}}\) is a weight module. Hence - \(\mathcal{M}^{\operatorname{ss}}\) is a weight module. Furthermore, given - \(\mu \in \lambda_k + Q\) + \ref{ex:quotient-is-weight-mod} that each quotient + \(\mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}\) is a weight + module. Hence \(\mathcal{M}^{\operatorname{ss}}\) is a weight module. + Furthermore, given \(\mu \in \mathfrak{h}^*\) \[ \mathcal{M}_\mu^{\operatorname{ss}} - = \bigoplus_{i j} + = \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}} \left( - \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}} + \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}} \right)_\mu - = \bigoplus_j + = \bigoplus_i \left( - \mfrac{\mathcal{M}_{k j + 1}}{\mathcal{M}_{k j}} + \mfrac{\mathcal{M}_{\mu i + 1}}{\mathcal{M}_{\mu i}} \right)_\mu - \cong \bigoplus_j - \mfrac{(\mathcal{M}_{k j + 1})_\mu} - {(\mathcal{M}_{k j})_\mu} + \cong \bigoplus_i + \mfrac{(\mathcal{M}_{\mu i + 1})_\mu} + {(\mathcal{M}_{\mu i})_\mu} \] In particular, \[ \dim \mathcal{M}_\mu^{\operatorname{ss}} - = \sum_j - \dim (\mathcal{M}_{k j + 1})_\mu - - \dim (\mathcal{M}_{k j})_\mu - = \dim \mathcal{M}[\lambda_k]_\mu + = \sum_i + \dim (\mathcal{M}_{\mu i + 1})_\mu - \dim (\mathcal{M}_{\mu i})_\mu + = \dim \mathcal{M}[\mu]_\mu = \dim \mathcal{M}_\mu = d \] @@ -621,20 +646,24 @@ named \emph{coherent families}. Likewise, given \(u \in \mathcal{U}(\mathfrak{g})_0\) the value \[ \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}}) - = \sum_j - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j + 1})_\mu}) - - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j})_\mu}) - = \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\lambda_k]_\mu}) + = \sum_i + \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{\mu i + 1})_\mu}) + - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{\mu i})_\mu}) + = \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\mu]_\mu}) = \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu}) \] is polynomial in \(\mu \in \mathfrak{h}^*\). \end{proof} -\begin{corollary}\label{thm:admissible-is-submod-of-extension} +As promised, if \(\mathcal{M}\) is a coherent extension of \(V\) then so is +\(\mathcal{M}^{\operatorname{ss}}\). + +\begin{proposition} 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 \(\mathcal{M}\). -\end{corollary} + \(\mathcal{M}\) 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 \(\mathcal{M}^{\operatorname{ss}}\). +\end{proposition} \begin{proof} Since \(V\) is irreducible, its support is contained in a single \(Q\)-coset. @@ -649,12 +678,70 @@ named \emph{coherent families}. \to \bigoplus_j \mfrac{\mathcal{M}_{j + 1}}{\mathcal{M}_j} \cong \mathcal{M}^{\operatorname{ss}}[\lambda] \] +\end{proof} + +Given the uniqueness of the semisimplification, the semisimplification of any +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 \(\mathcal{M}\). +\end{corollary} + +This last results provide a partial answer to the question of existence of nice +coherent extensions. A complementary question now is: wich submodules of a nice +coherent family are cuspidal representations? + +\begin{theorem}[Mathieu] + Let \(\mathcal{M}\) be a simple coherent family of degree \(d\) and + \(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent. + \begin{enumerate} + \item \(\mathcal{M}[\lambda]\) is irreducible. + \item \(F_\alpha\!\restriction_{\mathcal{M}[\lambda]}\) is injective for + all \(\alpha \in \Delta\). + \item \(\mathcal{M}[\lambda]\) is cuspidal. + \end{enumerate} +\end{theorem} + +\begin{proof} + The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly + from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the + corollary that \strong{(iii)} implies \strong{(ii)}. All it's 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 + \strong{(ii)}. We will show this is never the case.}. - It then follows from the uniqueness of the semisimplification of - \(\mathcal{M}\) that \(\mathcal{M} \cong \mathcal{M}^{\operatorname{ss}}\), - so we have an inclusion \(V \to \mathcal{M}\). + Suppose \(F_\alpha\) acts injectively in the subrepresentation + \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since + \(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains + an infinite-dimensiona 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 \(\mathcal{M}[\lambda] = V\). + + 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 \(U \cap \operatorname{supp}_{\operatorname{ess}} V\) + 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 \(\dim V_\mu = \deg V\). + + % Here we use thm:centralizer-multiplicity + In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any + irreducible \(\mathfrak{g}\)-module \(W\), the multiplicity of \(W\) in + \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) 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. \end{proof} +To finish the proof, we now show\dots + \begin{lemma} Let \(\mathcal{M}\) be a coherent family. The set \(U = \{\lambda \in \mathfrak{h}^* : \mathcal{M}_\lambda \ \text{is a simple @@ -775,51 +862,6 @@ named \emph{coherent families}. the union of Zariski-open subsets and is therefore open. We are done. \end{proof} -\begin{theorem}[Mathieu] - Let \(\mathcal{M}\) be a simple coherent family of degree \(d\) and - \(\lambda \in \mathfrak{h}^*\). The following conditions are equivalent. - \begin{enumerate} - \item \(\mathcal{M}[\lambda]\) is irreducible. - \item \(F_\alpha\!\restriction_{\mathcal{M}[\lambda]}\) is injective for - all \(\alpha \in \Delta\). - \item \(\mathcal{M}[\lambda]\) is cuspidal. - \end{enumerate} -\end{theorem} - -\begin{proof} - The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly - from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the - corollary that \strong{(iii)} implies \strong{(ii)}. All it's 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 - \strong{(ii)}. We will show this is never the case.}. - - Suppose \(F_\alpha\) acts injectively in the subrepresentation - \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since - \(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains - an infinite-dimensiona 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 claim - \(\mathcal{M}[\lambda] = V\). - - Since \(\mathcal{M}\) is simple and - \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense, \(U = \{\mu - \in \mathfrak{h}^* : \mathcal{M}_\mu \ \text{is a simple - $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is a non-empty open set, and \(U - \cap \operatorname{supp}_{\operatorname{ess}} V\) 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 \(\dim V_\mu = \deg - V\). - - In particular, \(V_\mu \ne 0\), so \(V_\mu = \mathcal{M}_\mu\). Now given any - irreducible \(\mathfrak{g}\)-module \(W\), the multiplicity of \(W\) in - \(\mathcal{M}[\lambda]\) is the same as the multiplicity \(W_\mu\) 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. -\end{proof} - \section{Localizations \& the Existance of Coherent Extensions} % TODO: Comment on the intuition behind the proof: we can get vectors in a