Generators and relations for the unitary group of a skew hermitian form over a local ring

TL;DR

This paper provides explicit generators and relations for the unitary group of a skew hermitian form over a local ring, and applies these results to construct Weil representations and prove surjectivity of reduction homomorphisms.

Contribution

It introduces a presentation of the unitary group using Bruhat generators and relations, and extends results on generation and surjectivity to broader settings.

Findings

Presented a Bruhat generator-based presentation of $U(2m,S)$.

Constructed an explicit Weil representation of $Sp(2m,R)$.

Proved surjectivity of reduction homomorphisms for $SU(2m,S)$ and $U(2m,S)$.

Abstract

Let $(S,*)$ be an involutive local ring and let $U(2m,S)$ be the unitary group associated to a nondegenerate skew hermitian form defined on a free $S$-module of rank $2m$. A presentation of $U(2m,S)$ is given in terms of Bruhat generators and their relations. This presentation is used to construct an explicit Weil representation of the symplectic group $Sp(2m,R)$ when $S=R$ is commutative and $*$ is the identity. When $S$ is commutative but $*$ is arbitrary with fixed ring $R$, an elementary proof that the special unitary group $SU(2m,S)$ is generated by unitary transvections is given. This is used to prove that the reduction homomorphisms $SU(2m,S)\to SU(2m,\tilde{S})$ and $U(2m,S)\to U(2m,\tilde{S})$ are surjective for any factor ring $\tilde{S}$ of $S$. The corresponding results for the symplectic group $Sp(2m,R)$ are obtained as corollaries when $*$ is the identity.