# Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences

## TL;DR

This paper presents a geometric approach to constructing Heisenberg-Weil representations for finite unitary groups and realizes the Howe correspondence geometrically, demonstrating unipotency preservation.

## Contribution

It introduces a novel geometric construction of Heisenberg-Weil representations and provides a geometric realization of the Howe correspondence over finite fields.

## Key findings

- Unipotency is preserved under the Howe correspondence.
- Provides a geometric construction using étale cohomology.
- Realizes Howe correspondence for $( ext{Sp}_{2n},O_2^-)$ over any finite field.

## Abstract

We give a geometric construction of the Heisenberg-Weil representation of a finite unitary group by the middle \'{e}tale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for $(\mathit{Sp}_{2n},O_2^-)$ over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.