The Taylor map on complex path groups

Abstract

Let W ( G ) denote the path group of an arbitrary complex connected Lie group. The existence of a heat kernel measure ν t on W ( G ) has been shown in [M. Cecil, B.K. Driver, Heat kernel measure on loop and path groups, preprint, http://www.math.uconn.edu/~cecil/papers/p2.pdf; Infin. Dimens. Anal. Quantum Probab. Relat. Top., submitted for publication]. The present work establishes an isometric map, the Taylor map, from the space of L 2 ( ν t ) -holomorphic functions on W ( G ) to a subspace of the dual of the universal enveloping algebra of Lie ( H ( G ) ) , where H ( G ) is the Lie subgroup of finite energy paths. This map is shown to be surjective in the case where G is a simply connected graded Lie group.