Extremal unipotent representations for the finite Howe correspondence

TL;DR

This paper investigates the extremal unipotent representations within the finite Howe correspondence for specific dual pairs over finite fields, identifying minimal and maximal irreducible subrepresentations.

Contribution

It introduces a method to extract extremal irreducible subrepresentations from unipotent representations in the finite Howe correspondence for classical groups.

Findings

Identified extremal subrepresentations in the Howe correspondence.

Provided a framework for analyzing unipotent representations.

Enhanced understanding of representation structure for dual pairs.

Abstract

We study the Howe correspondence for unipotent representations of irreducible dual pairs $(G',G)=(\text{U}_m(\mathbb{F}_q),\text{U}_n(\mathbb{F}_q))$ and $(G',G)=(\text{Sp}_{2m}(\mathbb{F}_q),\text{O}^\epsilon_{2n}(\mathbb{F}_q))$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd) and $\epsilon=\pm 1$. We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image of $\pi$ under the correpondence of a unipotent representation $\pi$ of $G$.