A Variant of the Induction Theorem for Springer Representations

Let G be a simple algebraic group over C with the Weyl group W . For a unipotent element u ∈ G, let Bu be the variety of Borel subgroups of G containing u. Let L be a Levi subgroup of a parabolic subgroup of G with the Weyl subgroup WL of W . Assume that u ∈ L and let B L u be a similar variety as Bu for L. We describe, for a certain choice of u ∈ L and e ≥ 1, the W -module ⊕ n≡k mod e H2n(Bu) for k = 0, . . . , e − 1, in terms of the WL-module H (B u ), which is a refinement of the induction theorem due to Lusztig. As an application, we give an explicit formula for the values of Green functions at root of unity, in the case where u is a regular unipotent element in L. 0. Introduction Let G be a connected reductive group over an algebraically closed field k, and W the Weyl group of G. For a unipotent element u ∈ G, let Bu be the variety of Borel subgroups containing u. According to Springer [Sp2], Lusztig [L1], W acts naturally on the l-adic cohomology group H(Bu) = H (Bu, Q̄l), the so-called Springer representations of W . Assume that k = C, or the characteristic p of k is good. Then it is known that H(Bu) = 0. We consider the graded W -module H(Bu) = ⊕ n≥0H (Bu). Let L be a Levi subgroup of a parabolic subgroup of G. Let WL be the Weyl group of L, which is naturally a subgroup of W . If u ∈ L, the variety B u is defined by replacing G by L, and we have a graded WL-module H(B u ). Lusztig proved in [L3] an induction theorem for Springer representations, which describes the W -module structure of H(Bu) in terms of the WL-module structure of H(B u ), in the case where u ∈ L. However in this theorem, the information on the graded W -module structure is eliminated. In this paper, we try to recover partly the graded W -module structure, i.e., for a fixed positive integer e, we consider the W -modules Ve,k = ⊕ n≡k mod e H (Bu) for k = 0, . . . , e−1. Let G be a simple group modulo center defined over C. We show, under a certain choice of u and e, that the W -module Ve,k can be described in terms of the graded WL-module H (B u ). In particular, we see that dim Ve,k is independent of the choice of k. ∗A part of this work was done during the author’s stay in CIB in Lausanne in June 2005. The author is grateful for their hospitality. 1