Weil representations via abstract data and Heisenberg groups: A comparison

Let B be a ring, not necessarily commutative, having an involution ⁎ and let U_2m(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 U_2m(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 U_2m(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 U_2m(B) generated by its Bruhat elements. Besides the independent interest, this subgroup and index are involved in the foregoing comparison of Weil representations.