Lehrer, G. I.; Shoji, T. On flag varieties, hyperplane complements and Springer representations of Weyl groups. (English) Zbl 0729.20017 J. Aust. Math. Soc., Ser. A 49, No. 3, 449-485 (1990). 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. Reviewer: V.Lakshmibai (Boston) Cited in 5 ReviewsCited in 8 Documents 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 Keywords:connected reductive group; Borel subgroup; Lie algebra; flag variety; action; Weyl group; multiplicity; Weyl group representations; Levi subalgebra; parabolic subalgebra; reflection representation PDFBibTeX XMLCite \textit{G. I. Lehrer} and \textit{T. Shoji}, J. Aust. Math. Soc., Ser. A 49, No. 3, 449--485 (1990; Zbl 0729.20017)