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Let $(M,J,\omega)$ be a quantizable compact K\"ahler manifold, withquantizing Hermitian line bundle $(A,h)$, and associated Hardy space $H(X)$,where $X$ is the unit circle bundle. Given a collection of $r$ Poissoncommuting quantizable Hamiltonian functions $f_j$ on $M$, there is an inducedAbelian unitary action on $H(X)$, generated by certain T\"oplitz operatorsnaturally induced by the $f_j$'s. As a multi-dimensional analogue of the usualWeyl law and trace formula, we consider the problem of describing theasymptotic clustering of the joint eigenvalues of these T\"oplitz operatorsalong a given ray, and locally on $M$ the asymptotic concentration of thecorresponding joint eigenfunctions. This problem naturally leads to a \lqdirectional local trace formula\rq, involving scaling asymptotics in theneighborhood of certain special loci in $M$. Under natural transversalityassumption, we obtain asymptotic expansions related to the local geometry ofthe Hamiltonian action and flow.

Local trace formulae for commuting Hamiltonians in T\"oplitz quantization