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( 1 T ∗Le(G), νe) ∩ D∞,p( 1 T ∗Le(G), νe) (1 < p < 2) be a closed 1-form on Le(G). Using results in rough path analysis, we prove that there exists a measurable function f on Le(G) such that df = α. Moreover we prove that dimker = 0 for the Hodge–Kodaira type operator acting on 1-forms on Le(G). © 2011 Elsevier Inc. All rights reserved.