Local trace formulae for commuting hamiltonians in toeplitz quantization

Let (M,J,ω) be a quantizable compact Kähler manifold, with quantizing Hermitian line bundle (A, h), and associated Hardy space H(X), where X is the unit circle bundle. Given a collection of r Poisson commuting quantizable Hamiltonian functions fj on M, there is an induced Abelian unitary action on H(X), generated by certain Toeplitz operators naturally induced by the fj’s. As a multi-dimensional analogue of the usual Weyl law and trace formula, we consider the problem of describing the asymptotic clustering of the joint eigenvalues of these Toeplitz operators along a given ray, and locally on M the asymptotic concentration of the corresponding joint eigenfunctions. This problem naturally leads to a ‘directional local trace formula’, involving scaling asymptotics in the neighborhood of certain special loci in M. Under natural transversality assumption, we obtain asymptotic expansions related to the local geometry of the Hamiltonian action and flow.