Weil representation of a generalized linear group over a ring of truncated polynomials (over a finite field

Luis Gutiérrez Frez, José Pantoja

Published 2015-06-26Version 1

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G=SL_*^{1}(2,A_n)$, ($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 $\rho$.