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

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

Projective completions of Jordan pairs Part II. Manifold structures and symmetric spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields \K, and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ``compact-like'' duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as "standard models" -- they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra

A Maslov cocycle for unitary groups

We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Chern class.Comment: To appear in Proc. London Math. So

Isotropy of unitary involutions

We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.Comment: final version, to appear in Acta Mat