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 at the identity and with time-interval [0, 1]. Then W(K ) forms an infinite-dimensional group under the operation of pointwise multiplication. Let \ denote the Wiener measure on W(K ). We construct an analog of the Segal Bargmann transform for W(K ). Let KC be the complexification of K, W(KC) the set of continuous paths in KC starting at the identity, and + the Wiener measure on W(KC). Our transform is a unitary map of L (W(K ), \) onto the ``holomorphic'' subspace of L(W(KC), +). By analogy with the classical transform, our transform is given by convolution with the Wiener measure, followed by analytic continuation. We prove that the transform for W(K ) is nicely related by means of the Itô map to the classical Segal Bargmann transform for the path-space in the Lie algebra of K. 1998 Academic Press