Extended Weil representations: the finite field cases

TL;DR

This paper explores the structure of Weil representations over finite fields, unifying different cases through twisted actions and extending the theory to even characteristic fields using geometric and lattice models.

Contribution

It introduces a unified approach to Weil representations over finite fields, including even characteristic cases, via twisted actions and geometric models, building on prior foundational works.

Findings

Reorganization of Weil representations as projective symplectic similitude group representations.

Extension of Weil representation theory to even characteristic finite fields.

Connection of lattice models with geometric Weil representations in characteristic two.

Abstract

It is well known(cf. Weil, G\'erardin's works) that there are two different Weil representations of a symplectic group over an odd finite field. By a twisted action, we show that one can reorganize them as a representation of a related projective symplectic similitude group. We also discuss the even field case by following Genestier-Lysenko and Gurevich-Hadani's works on geometric Weil representations in characteristic two. As a result, we approach some of their results from the lattice model, which is inspired by MVW, Prasad and Takeda's works.