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

Commit
4450a741f62295de32729f0b22922067259add37
Parent
d89fdc9c1c66a64e5dff3b79d344fdbc0683c97d
Author
Pablo <pablo-escobar@riseup.net>
Date

Corrected the statement of Fernando's theorem

Diffstat

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