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

Abstract

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 $(\Delta-\lambda_{z})^{\nu}$. 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 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.