Spectral decomposition and Siegel–Veech transforms for strata: the case of marked tori

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 Riemann surfaces. This space carries actions of the foliated Laplacian derived from the ${\mathrm{SL}}_2(\mathbb{R})$-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 continuous 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.