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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.

Camosso, S. (2016). Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions. ANNALI DI MATEMATICA PURA ED APPLICATA, 195(6), 2027-2059 [10.1007/s10231-016-0552-0].

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