An exact formula relating lattice points in symmetric spaces to the automorphic spectrum

Amy T. DeCelles
2012 Illinois Journal of Mathematics  
We extract an exact formula relating the number of lattice points in an expanding region of a complex semi-simple symmetric space and the automorphic spectrum from a spectral identity, which is obtained by producing two expressions for the automorphic fundamental solution of the invariant differential operator (∆ − λz) ν . On one hand, we form a Poincaré series from the solution to the corresponding differential equation on the free space G/K, which is obtained using the harmonic analysis of
more » ... onic analysis of bi-K-invariant functions. On the other hand, a suitable global automorphic Sobolev theory, developed in this paper, enables us to use the harmonic analysis of automorphic forms to produce a solution in terms of the automorphic spectrum.
doi:10.1215/ijm/1391178549 fatcat:pizpscyijrfnpeu7qduk2zd2ou