  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 $\rho$.

Representation Theory