Integration on loop groups. I. Quasi invariant measures

Abstract

On a riemannian manifold M, the Brownian motion defines a Wiener measure on the loops over M. In the case of R”, quasi invariance means the Cameron-Martin theorem [S] which fully describes the quasi invariance by translation of the Wiener measure. In order to state a generalization of this theorem we suppose that M is an homogeneous space under the action of a compact Lie group. The noncommutativity of this situation introduces a completely new phenomenon: the hypoellipticity associated to the Lie bracket of the basic vector fields. Curiously enough hypoellipticity, which helps to get regularity in finite dimensions, works against quasi invariance in infinite dimensions [18]. We denote by G a compact, connected Lie group, by H a connected subgroup of G, and by M the homogeneous space G/H. We denote by Y and 2’ the corresponding Lie algebras. We choose a euclidean metric on 9 which is Ad(g) invariant. This metric induces a riemannian metric on G and on M. We denote P,(G) the paths space on G, that is, the space of continuous maps y from [0, l] into G such that y(O) = e. We denote by [L,(G) the space of loops on G; that is,