Vanishing of one-dimensional L2-cohomologies of loop groups

Let G be a simply connected compact Lie group. Let Le(G) be the based loop group with the base point e which is the identity element. Let νe be the pinned Brownian motion measure on Le(G) and let α∈L2(⋀1T⁎Le(G),νe)∩D∞,p(⋀1T⁎Le(G),νe) (1