# Remarks on the theta correspondence over finite fields

## TL;DR

This paper investigates the theta correspondence over finite fields for symplectic-orthogonal dual pairs, utilizing Pan's decomposition of the Weil representation to relate unipotent and quadratic unipotent representations.

## Contribution

It applies Pan's decomposition to analyze the theta correspondence, providing new insights into the relationship between unipotent and quadratic unipotent representations over finite fields.

## Key findings

- Established the theta correspondence for unipotent and quadratic unipotent representations.
- Extended the approach of Adams-Moy and Aubert-Michel-Rouquier to finite field settings.
- Connected Pan's decomposition with the theta correspondence in this context.

## Abstract

S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.