Springer correspondence for the split symmetric pair in type $A$

TL;DR

This paper establishes the Springer correspondence for the symmetric pair (SL(N), SO(N)) using advanced geometric and algebraic tools, leading to new insights into Hessenberg varieties and Hecke algebra representations at q=-1.

Contribution

It introduces a novel Springer correspondence for the symmetric pair (SL(N), SO(N)) employing Fourier transform, parabolic induction, and nearby cycle sheaves.

Findings

Results on cohomology of Hessenberg varieties

Geometric constructions of irreducible Hecke algebra representations at q=-1

Extension of Springer correspondence to symmetric pairs

Abstract

In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{SL}(N),\mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain results on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.