Scaling asymptotics of Szegö kernels under commuting Hamiltonian actions

Abstract

Let M be a connected d M -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 d G , and let T be a compact torus of dimension d T . 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 S 1 -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 ν T for T , we consider the isotypical components associated to the multiples kν T , 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 ν G of G, we study the local scaling asymptotics of the equivariant Szegö projectors associated to ν G and kν T , for k → +∞, investigating their asymptotic concentration along certain loci defined by the moment maps.

Quantum Logic and Geometric Quantization

“…Let us denote with X the S 1 -bundle of L and with H(X) = L 2 (X) ∩ Ker ∂ b the Hardy space where ∂ b stands for the Cauchy-Riemann operator. We shall follow the scheme used in [17] under the action of a d G -dimensional compact Lie group G and a d T -dimensional torus T . We assume that these actions are Hamiltonian and holomorphic and that commute togheter.…”

“…In the same setting of [17], we have the action of the product group P = G×T on the symplectic manifold M. We shall interpret the von Neumann density operator as the equivariant Szegö projector Π. Now we spend few words on the Szegö projector.…”

“…In the paper [17] the main subject studied is a local asymptotics of the equivariant Szegö kernels Π ν G ,kν T , where the irreducible representation of T tends to infinity along a ray, and the irreducible representation of G is held fixed. The Szegö kernel is usually expressed in Heisenberg local coordinate centered at x ∈ X and for our purpose we shall need the scaling limits of Π ν G ,kν T on the diagonal of X × X.…”

“…, π : X → M is the canonical projection from the circle bundle to M, d T is the dimension of the torus, d G the dimension of the group G, det (C(m)) is a quantity associated to the metric and Φ G , Φ T are respectively the moment map of the group G and the torus T . Here we were under the assumptions that 0 ∈ g ∨ is a regular value for Φ G and 0 ∈ Φ T (for more datails see [17]).…”

Equivariant asymptotics of Szegö kernels under Hamiltonian $${{\varvec{U}}}(\mathbf{2})$$ U ( 2 ) -actions

“…In other words, we shall fix a ray in weight space and study the asymptotic behavior of the isotypes when the representation drifts to infinity along the ray. When G is a torus, this problem was studied in [6,26,27]; the case G = SU (2) is the object of [10]. To make this more precise, it is in order to set the geometric stage in detail.…”