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.

💡 Research Summary

The paper introduces a new 2‑cocycle, denoted μ, for symplectic and skew‑hermitian hyperbolic groups defined over arbitrary fields and division algebras equipped with an involution. The construction starts with a σ‑hermitian form h on a hyperbolic module V over a field F (or a skew‑field) together with its associated Lagrangian Grassmannian L(V). For any pair of Lagrangians L₁, L₂ ∈ L(V) the authors define a “Witt difference” by comparing the restrictions of h to the intersections and orthogonal complements of L₁ and L₂. This difference is an element of the Witt group W(F,σ) of σ‑hermitian forms, and the assignment (L₁, L₂) ↦ μ(L₁, L₂) satisfies the cocycle identity \