On flag varieties, hyperplane complements and Springer representations of Weyl groups
- 1 December 1990
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics
- Vol. 49 (3) , 449-485
- https://doi.org/10.1017/s1446788700032444
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 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.Keywords
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