# Weil representations via abstract data and Heisenberg groups: a comparison

## TL;DR

This paper constructs and compares Weil representations of unitary groups over rings with involution, using Heisenberg groups and axiomatic methods, providing explicit formulas, analyzing subgroup indices, and exploring implications for Gauss sums.

## Contribution

It introduces a new construction of Weil representations via Heisenberg groups for rings with involution and compares it with existing axiomatic approaches under general conditions.

## Key findings

- Explicit matrix form of Weil representations on Bruhat elements
- Comparison of two Weil representations under broad hypotheses
- Calculation of subgroup index generated by Bruhat elements in local rings

## Abstract

Let $B$ be a ring, not necessarily commutative, having an involution $*$ and let ${\mathrm U}_{2m}(B)$ 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 ${\mathrm U}_{2m}(B)$ 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 ${\mathrm U}_{2m}(B)$, 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 ${\mathrm U}_{2m}(B)$ generated by its Bruhat elements. Besides the independent interest, this subgroup and index are involved in the foregoing comparison of Weil representations.