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: 21b864b65b76846e3d18bdb8dba87a2cbc11e0dd
Parent: 12b23db03fe06c0261d967a8a2422e31a9cf9412
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 5 Oct 2022 12:05:14 +0000

Changed the notation for the Weyl group

Diffstat

2 files changed, 11 insertions, 11 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	2	1	1
Modified	sections/semisimple-algebras.tex	20	10	10

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -108,7 +108,7 @@
   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 \(w
-  \in W\).
+  \in \mathcal{W}\).
 \end{theorem}
 
 % TODO: Remark that the support of a simple weight module is always contained

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -2228,8 +2228,8 @@ is\dots
 \end{proof}
 
 \begin{definition}
-  We refer to the group \(W = \langle T_\alpha : \alpha \in \Delta^+ \rangle
-  \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl group of
+  We refer to the group \(\mathcal{W} = \langle T_\alpha : \alpha \in \Delta^+
+  \rangle \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl group of
   \(\mathfrak{g}\)}.
 \end{definition}
 
@@ -2243,7 +2243,7 @@ Indeed, the same argument leads us to the conclusion\dots
   highest weight \(\lambda\) are precisely the elements of the weight lattice
   \(P\) congruent to \(\lambda\) modulo the root lattice \(Q\) lying inside the
   convex hull of the image of \(\lambda\) under the action of the Weyl group
-  \(W\).
+  \(\mathcal{W}\).
 \end{theorem}
 
 Now the only thing we are missing for a complete classification is an existence
@@ -2488,13 +2488,13 @@ semisimple \(\mathfrak{g}\). Namely\dots
 The proof of proposition~\ref{thm:verma-is-finite-dim} is very technical and we
 won't include it here, but the idea behind it is to show that the set of
 weights of \(\sfrac{M(\lambda)}{N(\lambda)}\) is stable under the natural
-action of the Weyl group \(W\) in \(\mathfrak{h}^*\). One can then show that
-the every weight of \(V\) is conjugate to a single dominant integral weight of
-\(\sfrac{M(\lambda)}{N(\lambda)}\), and that the set of dominant integral
-weights of such irreducible quotient is finite. Since \(W\) is finitely
-generated, this implies the set of weights of the unique irreducible quotient
-of \(M(\lambda)\) is finite. But each weight space is finite-dimensional. Hence
-so is the irreducible quotient.
+action of the Weyl group \(\mathcal{W}\) in \(\mathfrak{h}^*\). One can then
+show that the every weight of \(V\) is conjugate to a single dominant integral
+weight of \(\sfrac{M(\lambda)}{N(\lambda)}\), and that the set of dominant
+integral weights of such irreducible quotient is finite. Since \(W\) is
+finitely generated, this implies the set of weights of the unique irreducible
+quotient of \(M(\lambda)\) is finite. But each weight space is
+finite-dimensional. Hence so is the irreducible quotient.
 
 We refer the reader to \cite[ch. 21]{humphreys} for further details. What we
 are really interested in is\dots