THE SEGAL-BARGMANN TRANSFORM FOR PATH-GROUPS

Let K be a connected Lie group of compact type and let W(K) denote the set of continuous paths in K, starting al the identity and with time- interval [0, 1]. Then W(K) forms an infinite-dimensional group under t he operation of pointwise multiplication. Let rho denote the Wiener me asure on W(K). We construct an analog of the Segal-Bargmann transform for W(K). Let K-C be the complexification of K, W(K-C) the set of cont inuous paths in K-C starting at the identity, and mu the Wiener measur e on W(K-C). Our transform is a unitary map of L-2(W(K), rho) onto the ''holomorphic'' subspace of L-2(W(K-C), mu). By analogy with the clas sical transform, our transform is given by convolution with the Wiener measure, followed by analytic continuation. We prove that the transfo rm for W(K) is nicely related by means of the Ito map to the classical Segal-Bargmann transform for the path-space in the Lie algebra of K. (C) 1998 Academic Press.