EXOTIC SYMMETRIC SPACES OF HIGHER LEVEL: SPRINGER CORRESPONDENCE FOR COMPLEX REFLECTION GROUPS

Abstract

Let G = GL(V) for a 2n-dimensional vector space V, and θ an involutive automorphism of G such that H = G ^θ ≃ Sp(V). Let be the set of unipotent elements g ∈ G such that θ(g) = g ⁻¹. For any integer r ≥ 2, we consider the variety , on which H acts diagonally. Let be a complex reflection group. In this paper, generalizing the known result for r = 2, we show that there exists a natural bijective correspondence (Springer correspondence) between the set of irreducible representations of W _n,r and a certain set of H-equivariant simple perverse sheaves on . We also consider a similar problem for , on which G acts diagonally, where G = GL(V) for a finite-dimensional vector space V.