Weil Representation of a Generalized Linear Group over a Ring of Truncated Polynomials over a Finite Field Endowed with a Second Class Involution

TL;DR

This paper constructs a Weil representation for a generalized linear group over a ring of truncated polynomials with an involution, providing a new decomposition and insights into its structure.

Contribution

It introduces a novel Weil representation for a specific generalized linear group over a ring with involution, including a decomposition and analysis of its unitary subgroup.

Findings

Constructed a Weil representation for G over a ring with involution

Provided a decomposition of the representation using a related unitary group

Described the structure of the associated unitary group

Abstract

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where $A_n$ is endowed with a second class involution. After the construction of a specific data, the representation is defined on the generators of a Bruhat presentation of $G$, via linear operators satisfying the relations of the presentation. The structure of a unitary group $U$ associated to $G$ is described. Using this group we obtain a first decomposition of $\rho$.