Abstract

Let B be a ring, not necessarily commutative, having an involution ⁎ and let U2m(B) be the unitary group of rank 2m associated to a hermitian or skew hermitian form relative to ⁎. When B is finite, we construct a Weil representation of U2m(B) via Heisenberg groups and find its explicit matrix form on the Bruhat elements. As a consequence, we derive information on generalized Gauss sums. On the other hand, there is an axiomatic method to define a Weil representation of U2m(B), and we compare the two Weil representations thus obtained under fairly general hypotheses. When B is local, not necessarily finite, we compute the index of the subgroup of U2m(B) generated by its Bruhat elements. Besides the independent interest, this subgroup and index are involved in the foregoing comparison of Weil representations.