  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.
