Asymptotics of Szegö kernels under Hamiltonian torus actions

Let X be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on X may be seen as a local manifestation of the decomposition of the (generalized) Hardy space H(X) into isotypes for the S 1-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of H(X) over the irreducible representations of the group. We focus here on the case of compact tori