A Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus

The group of area preserving diffeomorphisms of the annulus acts on its Lie algebra, the globally Hamiltonian vectorfields on the annulus. We consider a certain Hilbert space completion of this group (thinking of it as a group of unitary operators induced by the diffeomorphisms), and prove that the projection of an adjoint orbit onto a "Cartan" subalgebra isomorphic to Lz([0, 1]) is an infinitedimensional, weakly compact, convex set, whose extreme points coincide with the orbit, through a certain function, of the "permutation" semigroup of measure preserving transformations of [0, 1].