- Commit
- 4450a741f62295de32729f0b22922067259add37
- Parent
- d89fdc9c1c66a64e5dff3b79d344fdbc0683c97d
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Corrected the statement of Fernando's theorem
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Corrected the statement of Fernando's theorem
1 file changed, 28 insertions, 7 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/mathieu.tex | 35 | 28 | 7 |
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -120,21 +120,42 @@ which is \emph{not} parabolic induced. \end{definition} -% TODOOO: w should be an element of the subgroup W(V) of W. Fix this! % TODO: Remark on the fact that any simple weight p-mod is a (p/u)-mod, so that % the notation of a cuspidal p-mod is well definited -% TODO: Define the conjugation of a p-mod by an element of the Weil group +% TODO: Define the conjugation of a p-mod by an element of the Weil group? \begin{theorem}[Fernando] Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to \(L_{\mathfrak{p}}(V)\) for some parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) and some irreducible cuspidal \(\mathfrak{p}\)-module \(V\). - Furthermore, if \(\mathfrak{p}_1, \mathfrak{p}_2 \subset \mathfrak{g}\) are - parabolic and \(V_i\) is an irreducible cuspidal \(\mathfrak{p}_i\)-module - then \(L_{\mathfrak{p}_1}(V_1) \cong L_{\mathfrak{p}_2}(V_2)\) if, and only - if \(\mathfrak{p}_1 = \mathfrak{p}_2^w\) and \(V_1 \cong V_2^w\) for some \(w - \in \mathcal{W}\). \end{theorem} +% TODO: Point out that the relationship between p-modules and cuspidal +% g-modules is not 1-to-1 + +\begin{proposition}[Fernando] + Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\), there + exists a basis\footnote{This is usually called \emph{a $\mathfrak{p}$-adapted + basis}} \(\Sigma\) for \(\Delta\) such that \(\Sigma \subset + \Delta_{\mathfrak{p}_1}\). Furthermore, if \(\mathfrak{p}' \subset + \mathfrak{g}\) is another parabolic subalgebra, \(V\) is an irreducible + cuspidal \(\mathfrak{p}\)-module and \(W\) is an irreducible cuspidal + \(\mathfrak{p}'\)-module then \(L_{\mathfrak{p}}(V) \cong + L_{\mathfrak{p}'}(W)\) if, and only if \(\mathfrak{p}' = \mathfrak{p}^w\) and + \(W \cong V^w\) for some \(w \in \mathcal{W}_V\), where + \[ + \mathcal{W}_V + = \langle + T_\beta : \beta \in \Sigma, H_\beta + \mathfrak{u} + \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{u}} + \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ V + \rangle + \subset \mathcal{W} + \] +\end{proposition} + +% TODO: Point out that the definition of W_V is independant of the choice of +% Sigma + % TODO: Remark that the support of a simple weight module is always contained % in a coset % TODO: Note that conditions (ii), (iii) and (iv) have special names