The harmonic analysis of lattice counting on real spherical spaces

TL;DR

This paper extends the harmonic analysis approach to lattice counting from symmetric to real spherical spaces, aiming to deepen understanding of the interplay between arithmetic and randomness in these geometric contexts.

Contribution

It generalizes the lattice counting framework and harmonic analysis techniques from symmetric spaces to real spherical spaces, broadening the scope of the original approach.

Findings

Extended lattice counting methods to real spherical spaces.

Developed harmonic analysis tools for these spaces.

Enhanced understanding of arithmetic-randomness relationship.

Abstract

By the collective name of {\it lattice counting} we refer to a setup introduced in Duke-Rudnick-Sarnak that aim to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in Duke-Rudnick-Sarnak.