A generalized Weil representation for SL∗(2,Am), where Am=Fq[x]/〈xm〉

AbstractLet A be a unitary ring with an involution ∗. The groups SL∗(2,A), defined by Pantoja and Soto-Andrade in [J. Pantoja, J. Soto-Andrade, A Bruhat decomposition of the group SL∗(2,A), J. Algebra 262 (2003) 401–412], are a non-commutative version of the special linear groups SL(2,F) defined over a field F. Soto-Andrade, in [J. Soto-Andrade, Représentations de certains groupes symplectiques finis, Bull. Soc. Math. France Mém. 55–56 (1978)], using this approach gave a new presentation of the symplectic group Sp(2n,Fq) and used it to construct Weil representations of this group. In this paper, we classify the involutions of Am=Fq[x]/〈xm〉 and we study the groups SL∗(2,Am). We give a presentation of these groups as in [J. Pantoja, A presentation of the group SL∗(2,A), A a simple artinian ring with involution, Manuscripta Math. 121 (2006) 97–104] and finally, we construct a Weil representation of SL∗(2,Am) following the line of thought given in [J. Soto-Andrade, Représentations de certains groupes symplectiques finis, Bull. Soc. Math. France Mém. 55–56 (1978)]