Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Replaced the term "admissible" with the term "bounded"
The term "bounded" is apparently prefered in modern notation
The term "bounded" is certainly preferred by me
1 file changed, 27 insertions, 27 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/mathieu.tex | 54 | 27 | 27 |
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -120,10 +120,10 @@ to the case it holds. This brings us to the following definition. \end{example} A particularly well behaved class of examples are the so called -\emph{admissible} weight modules. +\emph{bounded} modules. -\begin{definition}\index{\(\mathfrak{g}\)-module!admissible modules}\index{\(\mathfrak{g}\)-module!(essential) support} - A weight \(\mathfrak{g}\)-module \(M\) is called \emph{admissible} if \(\dim +\begin{definition}\index{\(\mathfrak{g}\)-module!bounded modules}\index{\(\mathfrak{g}\)-module!(essential) support} + A weight \(\mathfrak{g}\)-module \(M\) is called \emph{bounded} 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 = @@ -136,7 +136,7 @@ A particularly well behaved class of examples are the so called (\ref{eq:laurent-polynomials-cusp-mod}). One can quickly verify \(K[x, x^{-1}]_{2 k} = K x^k\) and \(K[x, x^{-1}]_\lambda = 0\) for any \(\lambda \notin 2 \mathbb{Z}\), 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 + K x^k\) is a degree \(1\) bounded 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 submodule \(N \subset K[x, x^{-1}]\) must contain a monomial \(x^k\). But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} + @@ -164,13 +164,13 @@ M_{\lambda + \alpha}\) is stable under the action of \(\mathfrak{g}\) for all 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 \(\mathfrak{g}\)-module in the sense its essential support is +bounded \(\mathfrak{g}\)-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 \(M\) be an infinite-dimensional admissible + Let \(M\) be a simple infinite-dimensional bounded \(\mathfrak{g}\)-module. The essential support \(\operatorname{supp}_{\operatorname{ess}} M\) is Zariski-dense\footnote{Any choice of basis for $\mathfrak{h}^*$ induces a $K$-linear isomorphism @@ -421,7 +421,7 @@ so we can see \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}_{2 k + \frac{\lambda}{2}} = K x^k\) for all \(k \in \mathbb{Z}\) and \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}_\mu = 0\) for all other \(\mu \in \mathfrak{h}^*\). -Hence \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) is a degree \(1\) admissible +Hence \(\twisted{K[x, x^{-1}]}{\varphi_\lambda}\) is a degree \(1\) bounded \(\mathfrak{sl}_2(K)\)-module with \(\operatorname{supp} \twisted{K[x, x^{-1}]}{\varphi_\lambda} = \frac{\lambda}{2} + 2 \mathbb{Z}\). One can also quickly check that if \(\lambda \notin 1 + 2 \mathbb{Z}\) then \(e\) and \(f\) @@ -443,7 +443,7 @@ To a distracted spectator, \(\mathcal{M}\) may look like just another, innocent, \(\mathfrak{sl}_2(K)\)-module. However, the attentive reader may have already noticed some of the its bizarre features, most noticeable of which is the fact that \(\mathcal{M}\) is very big. In fact, \(\mathcal{M}\) is as big a -degree \(1\) admissible representation gets: \(\operatorname{supp} \mathcal{M} +degree \(1\) bounded module gets: \(\operatorname{supp} \mathcal{M} = \operatorname{supp}_{\operatorname{ess}} \mathcal{M}\) is the entirety of \(\mathfrak{h}^*\). This may look very alien the reader familiarized with the finite-dimensional setting, where the configuration of weights is very rigid. @@ -521,18 +521,18 @@ x^{-1}]\) and \(\mathcal{M}\) from Example~\ref{ex:sl-laurent-family}. This leads us to the following definition. \begin{definition}\index{coherent family!coherent extension} - Given an admissible \(\mathfrak{g}\)-module \(M\) of degree \(d\), a + Given a bounded \(\mathfrak{g}\)-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 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. +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. 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 @@ -670,7 +670,7 @@ promised, if \(\mathcal{M}\) is a coherent extension of \(M\) then so is \(\mathcal{M}^{\operatorname{ss}}\). \begin{proposition} - Let \(M\) be a simple admissible \(\mathfrak{g}\)-module and \(\mathcal{M}\) + Let \(M\) be a simple bounded \(\mathfrak{g}\)-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 \(\mathcal{M}^{\operatorname{ss}}\). @@ -695,8 +695,8 @@ Given the uniqueness of the semisimplification, the semisimplification of any semisimple coherent extension \(\mathcal{M}\) is \(\mathcal{M}\) itself and therefore\dots -\begin{corollary}\label{thm:admissible-is-submod-of-extension} - Let \(M\) be a simple admissible \(\mathfrak{g}\)-module and \(\mathcal{M}\) +\begin{corollary}\label{thm:bounded-is-submod-of-extension} + Let \(M\) be a simple bounded \(\mathfrak{g}\)-module and \(\mathcal{M}\) be a semisimple coherent extension of \(M\). Then \(M\) is contained in \(\mathcal{M}\). \end{corollary} @@ -893,7 +893,7 @@ coherent extensions, which will be the focus of our next section. \section{Localizations \& the Existence of Coherent Extensions} -Let \(M\) be a simple admissible \(\mathfrak{g}\)-module of degree \(d\). Our +Let \(M\) be a simple bounded \(\mathfrak{g}\)-module of degree \(d\). Our goal is to prove that \(M\) has a (unique) irreducible semisimple coherent extension \(\mathcal{M}\). Since \(M\) is simple, we know \(M \subset \mathcal{M}[\lambda]\) for any \(\lambda \in \operatorname{supp} M\). Our first @@ -1050,10 +1050,10 @@ 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 \(M\) be a simple infinite-dimensional admissible - \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots, - \beta_r\}\) for \(\Delta\) such that the elements \(F_{\beta_i}\) all act - injectively on \(M\) and satisfy \([F_{\beta_i}, F_{\beta_j}] = 0\). + Let \(M\) be a simple infinite-dimensional bounded \(\mathfrak{g}\)-module. + There is a basis \(\Sigma = \{\beta_1, \ldots, \beta_r\}\) 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} @@ -1082,8 +1082,8 @@ hypothesis of Lemma~\ref{thm:nice-basis-for-inversion}. We now show that \(\Sigma^{-1} M\) 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 \(\Sigma^{-1} M\) is an admissible +\begin{proposition}\label{thm:irr-bounded-is-contained-in-nice-mod} + The restriction of the localization \(\Sigma^{-1} M\) is a bounded \(\mathfrak{g}\)-module of degree \(d\) with \(\operatorname{supp} \Sigma^{-1} M = Q + \operatorname{supp} M\) and \(\dim \Sigma^{-1} M_\lambda = d\) for all \(\lambda \in \operatorname{supp} \Sigma^{-1} M\). @@ -1359,7 +1359,7 @@ Lo and behold\dots \(\mExt(M) = \mathcal{M}^{\operatorname{ss}}\). To see that \(\mExt(M)\) is irreducible, recall from - Corollary~\ref{thm:admissible-is-submod-of-extension} that \(M\) is a + Corollary~\ref{thm:bounded-is-submod-of-extension} that \(M\) is a \(\mathfrak{g}\)-submodule of \(\mExt(M)\). Since the degree of \(M\) is the same as the degree of \(\mExt(M)\), some of its weight spaces have maximal dimension inside of \(\mExt(M)\). In particular, it follows from