Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added notes on how coherent families are useful in other problems
Also removed some TODO items which were already dealt with
2 files changed, 8 insertions, 6 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Deleted | TODO.md | 6 | 0 | 6 |
| Modified | sections/mathieu.tex | 8 | 8 | 0 |
diff --git a/TODO.md b/TODO.md @@ -1,6 +0,0 @@ -# TODO - -* Comment on the fact that representation theory is larger than the - representation theory of Lie algebras -* Add some comments on how the concept of coherent families is useful to other - problems too
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -488,6 +488,14 @@ named \emph{coherent families}. K[x, x^{-1}]\). \end{example} +\begin{note} + We should point out that coherent families have proven themselves useful for + problems other than the classification of cuspidal \(\mathfrak{g}\)-modules. + For instance, Nilsson's classification of rank 1 \(\mathfrak{h}\)-free + representations of \(\mathfrak{sp}_{2 n}(K)\) is based on the notion of + coherent families and the so called \emph{weighting functor}. +\end{note} + 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