Projective unitary representations of infinite-dimensional Lie groups

Abstract

For an infinite-dimensional Lie group $G$ modeled on a locally convex Lie algebra $\mathfrak{g}$, 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 $\mathfrak{g}$, we characterize the unitary $\mathfrak{g}$-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 ${\mathfrak{g}}^{♯}$.