Hessenberg varieties, intersections of quadrics, and the Springer correspondence

TL;DR

This paper explores Hessenberg varieties linked to Springer theory for symmetric spaces, analyzing their geometry and monodromy representations through intersections of quadrics, and refines the Springer correspondence for a specific symmetric pair.

Contribution

It introduces a new class of Hessenberg varieties from Springer theory, studies their monodromy representations, and refines the Springer correspondence for (SL(N),SO(N)).

Findings

Decomposition of monodromy representations into irreducibles

Computation of Fourier transforms of IC complexes

Refinement of Springer correspondence for split symmetric pairs

Abstract

In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [CVX2].