Exotic symmetric spaces of higher level - Springer correspondence for complex reflection groups -

TL;DR

This paper extends the Springer correspondence to exotic symmetric spaces of higher level associated with complex reflection groups, establishing a bijection between irreducible representations and certain equivariant perverse sheaves.

Contribution

It generalizes the Springer correspondence for exotic symmetric spaces of level r, linking complex reflection group representations to geometric objects.

Findings

Established a bijective Springer correspondence for exotic symmetric spaces of level r.

Extended known results from the case r=2 to higher levels r > 1.

Connected irreducible representations of W_{n,r} with H-equivariant perverse sheaves.

Abstract

Let V be an 2n-dimensional vector space over an algebraically closed field of odd characteristic. Let G = GL(V), and H = Sp(V) the symplectic group contained in G. For a positive integer r > 1, we conisder the variety X = G/H \times V^{r-1}, on which H acts diagonally. X is called the exotic symmetric space of level r. Let W_{n,r} be the complex reflection group G(r,1,n). In this paper, generalizing the result for the case where 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 X_{uni}$, where X_{uni} is the "unipotent part" of X. We also consider a similar problem for G \times V^{r-1}, where G = GL(V) for a finite dimensional vector space V.