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 groupL(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.