arXiv subjects

Amy DeCelles

Publications and source records attributed to Amy DeCelles.

2 recordsLinked to original sources

arXiv2012

Fundamental solution for (Delta - lambda_z)^n on a symmetric space G/K

We determine a fundamental solution for the differential operator (Delta - lambda_z)^n on the Riemannian symmetric space G/K, where G is any complex semi-simple Lie group, and K is a maximal compact subgroup. We develop a global zonal spherical Sobolev theory, which enables us to use the harmonic analysis of spherical functions to obtain an integral representation for the solution. Then we obtain an explicit expression for the fundmantal solution, which allows relatively easy estimation of its behavior in the eigenvalue parameter lambda_z, with an eye towards further applications to automorphic forms involving asociated Poincare series.

math.RT↗

arXiv2011

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

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.

math.NT↗