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 → U ( n , C ) {S^1} \to U(n,{\mathbf {C}}) , extends to this thickened Grassmannian, and the measure is quasiinvariant with respect to these point transformations.