Degree Differences in the Eta Correspondences

TL;DR

This paper extends the eta-correspondence for finite reductive dual pairs beyond the stable range, providing a formula for the degree differences of paired irreducible characters.

Contribution

It generalizes the eta-correspondence to all finite reductive dual pairs and establishes a degree difference formula for paired characters.

Findings

Extended eta-correspondence to general finite reductive dual pairs

Determined the domain of the extended correspondence

Proved a formula for degree differences in q of paired characters

Abstract

A sub-relation of the $\Theta$-correspondence called the \emph{$\eta$-correspondence} is defined by Gurevich-Howe for a finite reductive dual pair in stable range. In this paper we propose an extension of the correspondence to general finite reductive dual pairs. Then we determine the domain the correspondence and prove a formula on the difference of degrees in $q$ of two irreducible characters paired by the correspondence.