  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)^N. On one hand, we form a Poincare
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.

Number Theory;Representation Theory