Non-existence of infinitesimally invariant measures on loop groups

Let G be a compact Lie group, L(G) the associated loop group, ω the canonical symplectic form on L(G). Set H the Hamiltonian function for which the associated ω-Hamiltonian vector field is the infinitesimal rotation. Then H generates a canonical semi-definite Riemannian structure on L(G), which induces a Riemannian structure on the free loop group L(G)/G = L0(G). This metric corresponds to the Sobolev norm H1. Using orthonormal frame methodology the positivity and finiteness of the Ricci curvature of L0(G) is proved. By studying the dissipation towards high modes of a unitary group valued SDE it is proved that the loop group does not have any infinitesimally invariant measure. © 2007 Elsevier Inc. All rights reserved.