Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions

Let M be a connected dM-dimensional complex projective manifold, and let A be a holomorphic positive Hermitian line bundle on M, with normalized curvature ω. Let G be a compact and connected Lie group of dimension dG, and let T be a compact torus of dimension dT. Suppose that both G and T act on M in a holomorphic and Hamiltonian manner, that the actions commute, and linearize to A. If X is the principal S1-bundle associated to A, then this setup determines commuting unitary representations of G and T on the Hardy space H(X) of X, which may then be decomposed over the irreducible representations of the two groups. If the moment map for the T-action is nowhere zero, all isotypical components for the torus are finite dimensional, and thus provide a collection of finite-dimensional G-modules. Given a nonzero integral weight νT for T, we consider the isotypical components associated to the multiples kνT, k→ + ∞, and focus on how their structure as G-modules is reflected by certain local scaling asymptotics on X (and M). More precisely, given a fixed irreducible character νG of G, we study the local scaling asymptotics of the equivariant Szegö projectors associated to νG and kνT, for k→ + ∞, investigating their asymptotic concentration along certain loci defined by the moment maps.