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On flag varieties, hyperplane complements and Springer representations of Weyl groups. (English) Zbl 0729.20017

Let G be a connected reductive group over \({\mathbb{C}}\). Let B be a Borel subgroup of G. Let \({\mathfrak g}\) be the Lie algebra of G, and let A be a nilpotent element of \({\mathfrak g}\). Let \({\mathcal B}_ A\) be the subvariety of the flag variety \({\mathcal B}(=G/B)\) defined by \({\mathcal B}_ A=\{gB\in {\mathcal B}|\) \(A\in Adg(Lie B)\}\). In this paper, the authors study the action of W (the Weyl group of G) on \(H^ i({\mathcal B}_ A)\), and prove an inequality for the multiplicity of the Weyl group representations which occur. In the case when A is a regular nilpotent element in a Levi subalgebra of a parabolic subalgebra of \({\mathfrak g}\), the authors determine the multiplicity of the reflection representation of W and show that this multiplicity is related to the geometry of the corresponding hyperplane complement. This paper makes a nice contribution to the geometry of the conjugacy classes.

MSC:

20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
20G10 Cohomology theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
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