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Szegö kernel equivariant asymptotics under Hamiltonian Lie group actions

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Abstract

Suppose that a compact and connected Lie group G acts on a complex Hodge manifold M in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle A on M. Then there is an induced unitary representation on the associated Hardy space and, if the moment map of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behavior of the corresponding equivariant Szegö kernels near certain loci defined by the moment map.

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Notes

  1. i.e., belonging to the open positive Weyl chamber

  2. We may interpret \(x+\mathbf {v}\) in terms of a system of Hesenberg local coordinates on X centered at x [17], smoothly varying with x, see §5.

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Acknowledgements

this study was supported by Universitá degli Studi di Milano-Bicocca (Grant No. 2017-ATE-0253). I am endebted to the referee for suggesting several improvements in presentation.

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  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 55, 20126, Milano, Italy

    Roberto Paoletti

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  1. Roberto Paoletti
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Correspondence to Roberto Paoletti.

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Paoletti, R. Szegö kernel equivariant asymptotics under Hamiltonian Lie group actions. J Geom Anal 32, 112 (2022). https://doi.org/10.1007/s12220-021-00829-4

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