J Austral Math Soc Ser A 49 pp449--485, 1990.

On Flag Varieties, Hyperplane Complements and Springer Representations of Weyl Groups

G. I. Lehrer and T. Shoji

Abstract

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 H¹(B_A) of the subvariety B_A (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.

1980 AMS Subject Classification (1985 Revision): 20G40

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Authors

G. I. Lehrer

University of Sydney, Sydney, NSW 2006, Australia.

T. Shoji

Science University of Tokyo, Noda Chiba 278, Tokyo, Japan.