Momentum Maps for Smooth Projective Unitary Representations

TL;DR

This paper explores the geometric structure of smooth projective unitary representations of Lie groups, demonstrating their Hamiltonian properties on associated Kähler manifolds and linking cocycles to momentum maps.

Contribution

It introduces the concept of momentum maps for these representations, showing their Hamiltonian nature and relating cocycles to the representation's structure.

Findings

The projective space of smooth vectors is a Kähler manifold.

The group action is weakly Hamiltonian and lifts to a Hamiltonian action.

Non-equivariance cocycles are identified with those from the representation.

Abstract

For a smooth projective unitary representation of a locally convex Lie group G, the projective space of smooth vectors is a locally convex Kaehler manifold. We show that the action of G on this space is weakly Hamiltonian, and lifts to a Hamiltonian action of the central U(1)-extension of G obtained from the projective representation. We identify the non-equivariance cocycles obtained from the weakly Hamiltonian action with those obtained from the projective representation, and give some integrality conditions on the image of the momentum map.