On invariant measures for the group of diffeomorphisms of the circle
  In a previous paper the author constructed biinvariant measures (possibly having values in a line bundle) for a loop group LK (with compact simply connected K) acting on the formal completion of its complexification LG. One motivation for this was to find a geometric construction for the unitary structure for the positive energy representations of LK. In this paper we pursue an analogous construction for the group of diffeomorphisms of the circle. 
Doug Pickrell