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: 96d61615e40b5e6b3251e7e14c6c72ecb489bd6a
Parent: 318630c4936a7f25e0fc287ea98505f31dc2b5c7
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 29 May 2023 01:18:45 +0000

Fixed another typo

Diffstat

1 file changed, 5 insertions, 5 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	10	5	5

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -897,11 +897,11 @@ family are cuspidal representations?
   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 \(\operatorname{supp}_{\operatorname{ess}} M\)
-  is Zariski-dense that \(U \cap \operatorname{supp}_{\operatorname{ess}} M\)
-  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 \(\dim M_\mu = \deg M\).
+  \(M\) is simple and \(\operatorname{supp}_{\operatorname{ess}} M\) is
+  Zariski-dense that \(U \cap \operatorname{supp}_{\operatorname{ess}} M\) 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
+  \(\dim M_\mu = \deg M\).
 
   In particular, \(M_\mu \ne 0\), so \(M_\mu = \mathcal{M}_\mu\). Now given any
   simple \(\mathfrak{g}\)-module \(L\), it follows from