Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Moved the part of the theorem on Ext about the central characters of the irreducible submodules of Ext to a separate theorem
1 files changed, 6 insertions, 3 deletions
| Status | Name | Changes | Insertions | Deletions |
| Modified | sections/mathieu.tex | 2 files changed | 6 | 3 |
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -306,9 +306,12 @@ \begin{theorem}[Mathieu] Let \(V\) be an infinite-dimensional admissible irreducible \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple - coherent extension \(\operatorname{Ext}(V)\) of \(V\). The central characters - of the irreducible submodules of \(\operatorname{Ext}(V)\) are all the same. - Furthermore, if \(\mathcal{M}\) is any coherent extension of \(V\), then + coherent extension \(\operatorname{Ext}(V)\) of \(V\). More precisely, if + \(\mathcal{M}\) is any coherent extension of \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\). \end{theorem} +\begin{proposition} + The central characters of the irreducible submodules of + \(\operatorname{Ext}(V)\) are all the same. +\end{proposition}