Mathematical Research Letters
Volume 24 (2017)
Counting Lattice Points in norm balls on higher rank simple Lie groups
Pages: 1285 – 1306
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.
simple Lie group, lattice points, spectral gap, spherical functions, Gelfand pairs, finite-dimensional representations, highest weight
2010 Mathematics Subject Classification
The first author acknowledges support of ERC grant 239606. The second author acknowledges support of ISF grant 2095/15.
Received 29 August 2016
Accepted 16 January 2017
Published 29 December 2017