On the reflection representation in Springer's theory
type: Item
instance of: scholarly article
title: On the reflection representation in Springer's theory
zbMATH Open document ID: 0823.20041
DOI: 10.1007/BF02566669
author: Nicolas Spaltenstein
published in: Commentarii Mathematici Helvetici
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/140252
review text: The theory of arrangements of hyperplanes allows to attach to every parabolic subgroup of a finite Coxeter group ``exponents of the Coxeter group''-like numbers. In the case of Weyl groups similar numbers arise from the theory of Springer's representations. \textit{G. I. Lehrer} and \textit{T. Shoji} [J. Aust. Math. Soc., Ser. A 49, 449-485 (1990; Zbl 0729.20017)] have shown that in characteristic zero these numbers coincide for classical Weyl groups. In the present paper, the complete determination of the multiplicity of the reflection representation in the Springer representations for classical groups with arbitrary characteristic is given. Both unipotent elements in the group and nilpotent elements in the Lie algebra are considered. Let \(\mathcal G\) be the Lie algebra of a connected reductive algebraic group defined over an algebraically closed field \(k\). Let \(\mathcal B\) be the variety of all Borel subgroups of \(G\). For \(A \in {\mathcal G}\) let \({\mathcal B}_ A = \{B \in {\mathcal B} \mid A \in \text{Lie} (B)\}\). The Springer representation for \(A\) is a natural action of the Weyl group \(W\) of \(G\) on the cohomology \(H^* ({\mathcal B}_ A)\). The cohomology theory used is \(l\)-adic cohomology. Let \(\rho\) be the reflection representation of \(W\). In the present paper, the multiplicity of \(\rho\) in the cohomology groups \(H^ i ({\mathcal B}_ A)\) (\(A\) nilpotent, \(G\) a classical group) is determined. Let \(\mathcal P\) be a conjugacy class of parabolic subgroups of \(G\), and for \(A \in {\mathcal G}\), let \({\mathcal P}_ A = \{P \in {\mathcal P} \mid A \in \text{Lie} (P)\}\). Also, let \(W({\mathcal P})\) be a subgroup of \(W\) which can be thought of as the Weyl group of \(P \in {\mathcal P}\). The following theorem due to Borho-MacPherson is essential; Theorem. The homomorphism \(\pi^*_ A: H^*({\mathcal P}_ A) \to H^*({\mathcal B}_ A)\) induces an isomorphism of \(W^ P\)-modules \(H^*({\mathcal P}_ A) \cong H^* ({\mathcal B}_ A)^{W(P)}…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 57100
zbMATH Keywords: multiplicity of reflection representation
zbMATH Keywords: arrangements of hyperplanes
zbMATH Keywords: finite Coxeter groups
zbMATH Keywords: Weyl groups
zbMATH Keywords: Springer representations
zbMATH Keywords: classical groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: connected reductive algebraic groups
zbMATH Keywords: Borel subgroups
zbMATH Keywords: \(l\)-adic cohomology
zbMATH Keywords: cohomology groups
zbMATH Keywords: parabolic subgroups
reviewed by: Yasuhiro Asoo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 57100