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

Abstract

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\mathrm{SL}}_{\ast }^1(2,A_n)$ ($A_n=K[x]/⟨x^n⟩$, $K$ 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 $\rho$.

References in corpus (3)

• On character values and decomposition of the Weil representation associated to a finite abelian group
• Combinatorics of finite abelian groups and Weil representations
• A Generalized Weil Representation for the finite split orthogonal group $O_q(2n,2n)$, $q$ odd greater than $3$