The light cone Siegel transform, its moment formulas and applications

The classical Siegel transform is a transform which takes functions on the Euclidean space to functions on the space of lattices.

In this talk I will discuss a new type of Siegel transform where the role of the Euclidean space is replaced by the light cone of a certain indefinite integral quadratic form. In this setting one can use the spectral theory of incomplete Eisenstein series to prove explicit first and second moment formulas for this transform, generalizing the classical results of Siegel and Rogers. I'll also discuss some applications of our moment formula to various counting problems, including one on intrinsic Diophantine approximations on spheres. This is work in progress with Dubi Kelmer.