The heat kernel measure $\nu_}t}$ is constructed on $\ mathcal}W}(G),$ the group of paths based at the identity \ on a simply connected complex Lie group $G.$ \ An isometric map, the Taylor map, is established from the space of $L̂}2}(\nu_}t})-$holomorphic functions on $\ mathcal}W}(G)$ 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. \ Surjectivity of this Taylor map can be shown in the case where $G$ is stratified nilpotent