On the reflection representation in Springer's theory
The theory of arrangements of hyperplanes allows to attach to every parabolic subgroup of a finite Coxeter group ``exponents of the Coxeter group''-like numbers. In the case of Weyl groups similar numbers arise from the theory of Springer's representations. \textit{G. I. Lehrer} and \textit{T. Shoji} [J. Aust. Math. Soc., Ser. A 49, 449-485 (1990; Zbl 0729.20017)] have shown that in characteristic zero these numbers coincide for classical Weyl groups. In the present paper, the complete determination of the multiplicity of the reflection representation in the Springer representations for classical groups with arbitrary characteristic is given. Both unipotent elements in the group and nilpotent elements in the Lie algebra are considered. Let \(\mathcal G\) be the Lie algebra of a connected reductive algebraic group defined over an algebraically closed field \(k\). Let \(\mathcal B\) be the variety of all Borel subgroups of \(G\). For \(A \in {\mathcal G}\) let \({\mathcal B}_ A = \{B \in {\mathcal B} \mid A \in \text{Lie} (B)\}\). The Springer representation for \(A\) is a natural action of the Weyl group \(W\) of \(G\) on the cohomology \(H^* ({\mathcal B}_ A)\). The cohomology theory used is \(l\)-adic cohomology. Let \(\rho\) be the reflection representation of \(W\). In the present paper, the multiplicity of \(\rho\) in the cohomology groups \(H^ i ({\mathcal B}_ A)\) (\(A\) nilpotent, \(G\) a classical group) is determined. Let \(\mathcal P\) be a conjugacy class of parabolic subgroups of \(G\), and for \(A \in {\mathcal G}\), let \({\mathcal P}_ A = \{P \in {\mathcal P} \mid A \in \text{Lie} (P)\}\). Also, let \(W({\mathcal P})\) be a subgroup of \(W\) which can be thought of as the Weyl group of \(P \in {\mathcal P}\). The following theorem due to Borho-MacPherson is essential; Theorem. The homomorphism \(\pi^*_ A: H^*({\mathcal P}_ A) \to H^*({\mathcal B}_ A)\) induces an isomorphism of \(W^ P\)-modules \(H^*({\mathcal P}_ A) \cong H^* ({\mathcal B}_ A)^{W(P)}\), where \(W^ P = N(P)/W(P)\), \(N(P) = {\mathcal N}_ W (W({\mathcal P}))\). A proof of this theorem is outlined in the present paper. In the case of \(\text{GL}_ n\), \(\text{Sp}_{2n}\) and \(\text{SO}_{2n + 1}\), results of Lehrer and Shoji for characteristic zero are reproved and also proved for arbitrary characteristic. In the case of \(\text{SO}_{2n}\) and for arbitrary characteristic, the same results are proved.
multiplicity of reflection representation
scientific article; zbMATH DE number 57100