A Maslov cocycle for unitary groups

TL;DR

This paper introduces a universal 2-cocycle for symplectic and skew-hermitian hyperbolic groups over various fields, generalizing the first Chern class and exhibiting key functorial properties.

Contribution

It constructs a new 2-cocycle with functorial and stability properties, extending classical characteristic classes to broader algebraic groups.

Findings

Defines a 2-cocycle valued in the Witt group

Shows the cocycle coincides with the first Chern class over R and C

Establishes functoriality and stability of the cocycle

Abstract

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.