Exotic symmetric space over a finite field, I

TL;DR

This paper develops a theory of character sheaves on the exotic symmetric space over a finite field and provides an alternative proof of the Springer correspondence connecting irreducible characters of the Weyl group to H-orbits.

Contribution

It introduces a new framework of character sheaves on the exotic symmetric space and offers an alternative proof of the Springer correspondence using this theory.

Findings

Established a theory of character sheaves on the exotic symmetric space.

Provided an alternative proof of the Springer correspondence.

Linked irreducible characters of the Weyl group to H-orbits via character sheaves.

Abstract

Let V be a 2n dimensional vector space over an algebraically closed field of odd characteristic. Let G = GL(V) and H = Sp_{2n} be the symplectic group contained in G. We call X = G/H \times V the exotic symmetric space, since its "unipotent" part is isomorphic to Kato's exotic nilcone. Let W be the Weyl group of type C_n. Kato established the Springer correspondence between irreducible characters of W and H-orbits in the exotic nilcone, based on the Ginzburg theory of affine Hecke algebras. In this paper, we develpoe a theory of character sheaves on X, and give an alternate proof for the Springer correspondence based on the theroy of character sheaves.