Theta correspondence and Springer correspondence

TL;DR

This paper derives explicit formulas connecting theta and Springer correspondences for unipotent principal-series representations over finite fields, advancing understanding of their structure and multiplicities.

Contribution

It provides a new explicit formula linking theta correspondence with Springer correspondence and extends results to module categories of Hecke categories from spherical varieties.

Findings

Explicit formula for theta correspondence in terms of Springer correspondence

General results on module categories of Hecke categories

Formula for multiplicities of unipotent principal series representations

Abstract

In this paper, we obtain an explicit formula for the theta correspondence of unipotent principal-series representations between an even orthogonal and a symplectic group or between general linear groups over a finite field. The formula is in terms of the Springer correspondence. Along the way we prove general results about module categories of Hecke categories arising from spherical varieties, and give a similar formula for the multiplicities of the unipotent principal series representations in the function space of the spherical variety in terms of relative Springer theory.