Vanishing of one dimensional L^2-cohomologies of loop groups

Let $G$ be a simply connected compact Lie group. Let $L_e(G)$ be the based loop group with the base point $e$ which is the identity element. Let $\nu_e$ be the pinned Brownian motion measure on $L_e(G)$ and let $\alpha\in L^2(\wedge^1T^{\ast}L_e(G),\nu_e)\cap {\mathbb D}^{\infty,p}(\wedge^1T^{\ast}L_e(G),\nu_e)$ $(1<p<2)$ be a closed 1-form on $L_e(G)$. Using results in rough path analysis, we prove that there exists a measurable function $f$ on $L_e(G)$ such that $df=\alpha$. Moreover we prove that $\dim\ker \square=0$ for the Hodge-Kodaira type operator $\square$ acting on 1-forms on $L_e(G)$.