A generalized Weil representation for \(\text{SL}_*(2,A_m)\), where \(A_m=\mathbb{F}_q[x]\langle x^m\rangle\). (Q837037)

Let \(k\) be a finite field of odd order, \(m\) a positive integer, \(A_m=k[x]/(x^m)\). The involution \(^*\) on \(A_m\) is defined by \(x^*=-x\). The other involutions are classified. A presentation of the unitary group \(\text{SL}_*(2,A_m)\) of the standard skew-Hermitian matrix \(\left(\begin{smallmatrix} 0&1\\ -1& 0\end{smallmatrix}\right)\) is given, and a Weil representation of this group is constructed.