On the Howe correspondence and Harish-Chandra series

TL;DR

This paper investigates how the Howe correspondence influences Harish-Chandra series for certain dual pairs over finite fields, using Lusztig correspondence to simplify the analysis and providing two applications of the findings.

Contribution

It introduces a method to analyze the Howe correspondence's effect on Harish-Chandra series via Lusztig correspondence, focusing on unipotent series for specific dual pairs.

Findings

Reduction of Howe correspondence analysis to unipotent Harish-Chandra series

Application of Lusztig correspondence in this context

Two new applications derived from the theoretical results

Abstract

We study the effect of the Howe correspondence on Harish-Chandra series for type I dual pairs ($\mathbf{U}_m(\mathbb{F}_q),\mathbf{U}_n(\mathbb{F}_q$)) and ($\mathbf{Sp}_{2m}(\mathbb{F}_q),\mathbf{O}^\pm_{2n}(\mathbb{F}_q)$), where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd). We use the Lusztig correspondence to reduce the study of the Howe correspondence between Harish-Chandra series to a correspondence between unipotent Harish-Chandra series. Finally, we provide two applications of these results.