On the Support of Quasi-Invariant Measures on Infinite-Dimensional Grassmann Manifolds

Abstract

One antisymmetric analogue of Gaussian measure on a Hilbert space is a certain measure on an infinite-dimensional Grassmann manifold. The main purpose of this paper is to show that the characteristic function of this measure is continuous in a weighted norm for graph coordinates. As a consequence the measure is supported on a thickened Grassmann manifold. The action of certain unitary transformations, in particular smooth loops ${S^1} \to U(n,{\mathbf {C}})$, extends to this thickened Grassmannian, and the measure is quasiinvariant with respect to these point transformations.