Projective unitary representations of infinite-dimensional Lie groups

Abstract

For an infinite-dimensional Lie group G modeled on a locally convex Lie algebra g, we prove that every smooth projective unitary representation of G corresponds to a smooth linear unitary representation of a Lie group extension G♯ of G. (The main point is the smooth structure on G♯.) For infinite-dimensional Lie groups G which are 1-connected, regular, and modeled on a barreled Lie algebra g, we characterize the unitary g-representations which integrate to G. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of G, smooth linear unitary representations of G♯, and the appropriate unitary representations of its Lie algebra g♯.