Exotic symmetric space over a finite field, I

Let V be a 2n-dimensional vector space over an algebraically closed field k with ch k ≠ 2. Let G = GL(V) and H = Sp2n be the symplectic group obtained as H = Gθ for an involution θ on G. We also denote by θ the induced involution on $ \mathfrak{g} $ = Lie G. Consider the variety G/H × V on which H acts naturally. Let $ \mathfrak{g}_{\mathrm{nil}}^{{-\theta }} $ be the set of nilpotent elements in the -1 eigenspace of θ in $ \mathfrak{g} $. The role of the unipotent variety for G in our setup is played by $ \mathfrak{g}_{\mathrm{nil}}^{{-\theta }} $ × V, which coincides with Kato’s exotic nilpotent cone. Kato established, in the case where k = C, the Springer correspondence between the set of irreducible representations of the Weyl group of type Cn and the set of H-orbits in $ \mathfrak{g}_{\mathrm{nil}}^{{-\theta }} $ × V by applying Ginzburg theory for affine Hecke algebras. In this paper we develop a theory of character sheaves on G/H × V, and give an alternate proof for Kato’s result on the Springer correspondence based on the theory of character sheaves.