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On flag varieties, hyperplane complements and Springer representations of Weyl groups

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  09 April 2009

G. I. Lehrer  and
T. Shoji
G. I. Lehrer
Affiliation:
University of SydneySydney, NSW 2006, Australia
T. Shoji
Affiliation:
Science University of TokyoNoda Chiba 278 Tokyo, Japan
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Abstract

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Let G be a connected reductive linear algebraic group over the complex numbers. For any element A of the Lie algebra of G, there is an action of the Weyl group W on the cohomology Hi(BA) of the subvariety BA (see below for the definition) of the flag variety of G. We study this action and prove an inequality for the multiplicity of the Weyl group representations which occur ((4.8) below). This involves geometric data. This inequality is applied to determine the multiplicity of the reflection representation of W when A is a nilpotent element of “parabolic type”. In particular this multiplicity is related to the geometry of the corresponding hyperplane complement.

MSC classification

Secondary: 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 3 , December 1990 , pp. 449 - 485
Copyright
Copyright © Australian Mathematical Society 1990

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