lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 3ae1a99de762cef0a6de84712b80254f81c71bdd
Parent: 0431480fc81638aae0556e1c8461b2e83607a9a0
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 5 Oct 2022 19:13:13 +0000

Added a proof of the fact that the localization of an admissible module is well behaved

Diffstat

1 file changed, 69 insertions, 3 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	72	69	3

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -295,11 +295,77 @@
 
 \begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
   Let \(V\) be an irreducible infinite-dimensional admissible
-  \(\mathfrak{g}\)-module. Then \(V\) is contained in a weight module \(W\) of
-  degree \(d\) such that \(\operatorname{supp} W = Q + \operatorname{supp} V\)
-  and \(\dim W_\lambda = d\) for all \(\lambda \in \operatorname{supp} W\).
+  \(\mathfrak{g}\)-module. Then \(V\) is contained in a
+  \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\)-module \(W\),
+  whose restriction to the scalars in \(\mathcal{U}(\mathfrak{g})\) is a weight
+  module of degree \(d\) such that \(\operatorname{supp} W = Q +
+  \operatorname{supp} V\) and \(\dim W_\lambda = d\) for all \(\lambda \in
+  \operatorname{supp} W\).
 \end{proposition}
 
+\begin{proof}
+  Take \(W = \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
+  \otimes_{\mathcal{U}(\mathfrak{g})} V\). Since each \(F_\beta\) acts
+  injectively in \(V\), the localization map
+  \begin{align*}
+    V & \to     W           \\
+    v & \mapsto 1 \otimes v
+  \end{align*}
+  is injective. In particular, we may regard \(V\) as a
+  \(\mathfrak{g}\)-submodule of \(W\).
+
+  Fix some \(\beta \in \Sigma\). We begin by show that \(F_\beta\) and
+  \(F_\beta^{-1}\) map the weight space \(V_\lambda\) to the weight spaces
+  \(W_{\lambda - \beta}\) and \(W_{\lambda + \beta}\) respectively. Indeed,
+  given \(v \in V_\lambda\) and \(H \in \mathfrak{h}\) we have
+  \[
+    H F_\beta v
+    = ([H, F_\beta] + F_\beta H)v
+    = F_\beta (-\beta(H) + H) v
+    = F_\beta (\lambda - \beta)(H) \cdot v
+    = (\lambda - \beta)(H) \cdot F_\beta v
+  \]
+
+  On the other hand,
+  \[
+    0
+    = [H, 1]
+    = [H, F_\beta F_\beta^{-1}]
+    = F_\beta [H, F_\beta^{-1}] + [H, F_\beta] F_\beta^{-1}
+    = F_\beta [H, F_\beta^{-1}] - \beta(H) F_\beta F_\beta^{-1},
+  \]
+  so that \([H, F_\beta^{-1}] = \beta(H) \cdot F_\beta^{-1}\) and therefore
+  \[
+    H F_\beta^{-1} v
+    = ([H, F_\beta^{-1}] + F_\beta^{-1} H) v
+    = F_\beta^{-1} (\beta(H) + H) v
+    = F_\beta^{-1} (\lambda + \beta)(H) \cdot v
+  \]
+
+  % TODO: Remark beforehand that any element of the localization of V may be
+  % written as an element of v tensored by an element of the form 1/s
+  From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(W_{\lambda \pm
+  \beta}\) follows our first conclusion: since \(V\) is a weight module and
+  every element of \(W\) 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 \(W =
+  \bigoplus_\lambda W_\lambda\). Furtheremore, since the action of each
+  \(F_\beta\) in \(W\) is bijective and \(\Sigma\) is a basis of \(Q\) we
+  obtain \(\operatorname{supp} W = Q + \operatorname{supp} V\). 
+
+  % TODOOOOOO: Change the notation for the subspace U
+  Again, because of the bijectivity of the \(F_\beta\)'s, to see that \(\dim
+  W_\lambda = d\) for all \(\lambda \in \mathfrak{h}^*\) it suffices to show
+  that \(\dim W_\lambda = d\) for some \(\lambda \in \operatorname{supp} W\).
+  We may take \(\lambda \operatorname{supp} V\) with \(\dim V_\lambda = d\).
+  For any finite-dimensional subspace \(U \subset W_\lambda\) we can find \(s
+  \in (F_\beta)_{\beta \in \Sigma}\) such that \(s U \subset V\). If \(s =
+  F_{\beta_{i_1}} \cdots F_{\beta_{i_n}}\), it is clear \(s U \subset
+  V_{\lambda - \beta_{i_1} - \cdots - \beta_{i_n}}\), so \(\dim U \le d\) --
+  \(s\) is injective. This holds for all finite-dimensional \(U \subset
+  W_\lambda\), so \(\dim W_\lambda \le d\). It then follows from the fact that
+  \(V_\lambda \subset W_\lambda\) that \(\dim W_\lambda = d\).
+\end{proof}
+
 % TODO: Remark that any module over the localization is a g-module if we
 % restrict it via the localization map, wich is injective in this case
 \begin{proposition}\label{thm:nice-automorphisms-exist}