On Springer's representations of Weyl groups containing -1
type: Item
instance of: scholarly article
title: On Springer's representations of Weyl groups containing -1
zbMATH Open document ID: 0548.20026
DOI: 10.1007/BF01193777
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let G be a connected reductive complex algebraic group. For \(u\in G\) unipotent, let \({\mathcal B}_ u\) be the variety of all Borel subgroups of G containing x, \(\beta\) (u) its dimension. Springer has defined a natural representation of the Weyl group W of G in \(H^*({\mathcal B}_ u;{\mathbb{C}})\) which commutes with the natural action of the finite group \(A(u)=C_ G(u)/C^ 0_ G(u)\). Let \(\rho\),\(\phi\) be irreducible representations of W, A(u) respectively. In the case where -1\(\in W\), the paper gives a necessary condition mod 2 on \(i\in {\mathbb{N}}\) for \(\rho \otimes\phi \) to occur in \(H^{2i}({\mathcal B}_ u;{\mathbb{C}})\) (it is known that \(H^{2i- 1}({\mathcal B}_ u;{\mathbb{C}})=0)\). The proof uses results of the representation theory of finite Chevalley groups, in particular an algorithm discovered by Shoji to compute the Green functions of finite Chevalley groups.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 3873593
zbMATH Keywords: connected reductive complex algebraic group
zbMATH Keywords: Borel subgroups
zbMATH Keywords: irreducible representations
zbMATH Keywords: Green functions of finite Chevalley groups
author: Nicolas Spaltenstein
MaRDI profile type: Publication
cites work: Green functions of finite Chevalley groups of type \(E_ n (n=6,7,8)\)
cites work: Q3919888
cites work: On the green polynomials of a chevalley group of type f4
cites work: On the Green polynomials of classical groups
cites work: Trigonometric sums, Green functions of finite groups and representations of Weyl groups
cites work: Green polynomials op finite classical groups
full work available at URL: https://doi.org/10.1007/bf01193777
OpenAlex ID: W2087082564
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description: scientific article; zbMATH DE number 3873593