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

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)→SU(2m,S˜) and U(2m,S)→U(2m,S˜) are surjective for any factor ring S˜ of S. The corresponding results for the symplectic group Sp(2m,R) are obtained as corollaries when ⁎ is the identity.