Weil representations of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$, $q>3$ odd via presentation and compatibility of methods

TL;DR

This paper constructs Weil representations of quasi-split unitary groups over finite fields using a presentation approach and examines their compatibility with classical constructions.

Contribution

It introduces a novel presentation-based method for constructing Weil representations of $U(n,n)$ over finite fields and analyzes their compatibility with existing classical representations.

Findings

Weil representations are explicitly constructed via group presentations.

The compatibility with Gérardin's classical representations is established.

The approach provides a new perspective on representations of unitary groups over finite fields.

Abstract

In this article we construct Weil representations of quasi-split unitary groups $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ associated to quadratic extensions of finite fields. We define these representations by using an adequate presentation Bruhat like of those groups. More precisely, we define Weil representations of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ associating to each generator a linear map of a suitable $\mathbb{C}$-vector space satisfying the relations of the aforementioned presentation. In addition, we also address the natural question on the compatibility of our representation of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ with the classical one constructed by G\'erardin.