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arXiv · 2404.06597

Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori

Jayadev Athreya·Jean Lagacé·Martin Möller·Martin Raum
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Abstract

Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections. We also show that the contiunuous spectrum of the foliated Laplacian is much larger than the space of Siegel-Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations.

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Jayadev Athreya, Jean Lagacé, Martin Möller, Martin Raum. 2024-04-09. Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori. https://arxiv.org/abs/2404.06597

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