A Generalized Weil Representation for the finite split orthogonal group $O_q(2n,2n)$, $q$ odd greater than $3$

TL;DR

This paper constructs a generalized Weil representation for the split orthogonal group over finite fields with odd q, providing a new perspective on its structure and relation to symplectic groups.

Contribution

It introduces a generalized Weil representation for $ ext{O}_q(2n,2n)$ using generators and relations, and relates it to the classical Weil representation via restriction.

Findings

Representation constructed via generators and relations

Initial decomposition of the representation provided

Representation matches restriction from classical Weil representation

Abstract

We construct via generators and relations a generalized Weil representation for the split orthogonal group $\ot_q(2n,2n)$ over a finite field of $q$ elements. Besides, we give an initial decomposition of the representation found. We also show that the constructed representation is equal to the restriction of the Weil representation to $\ot_q(2n,2n)$ for the reductive dual pair $(\Sp_2(\mathbb{F}_q),\ot_q(2n,2n))$ and that the initial decomposition is the same as the decomposition with respect to the action of $\Sp_2(\mathbb{F}_q)$.