Mathematical Research Letters

Volume 24 (2017)

Number 5

Counting Lattice Points in norm balls on higher rank simple Lie groups

Pages: 1285 – 1306

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n5.a3

Authors

Alexander Gorodnik (School of Mathematics, University of Bristol, United Kingdom)

Amos Nevo (Department of Mathematics, Technion IIT, Haifa, Israel)

Gal Yehoshua (Department of Mathematics, Technion IIT, Haifa, Israel)

Abstract

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. We focus particularly on the case of the special linear groups where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly twice as good in some cases.

Keywords

simple Lie group, lattice points, spectral gap, spherical functions, Gelfand pairs, finite-dimensional representations, highest weight

2010 Mathematics Subject Classification

11K60, 37A17

Full Text (PDF format)

The first author acknowledges support of ERC grant 239606. The second author acknowledges support of ISF grant 2095/15.

Paper received on 29 August 2016.