Resolvent and lattice points on symmetric spaces of strictly negative curvature

Abstract.

We study the asymptotics of the lattice point counting function $N(x,y;r)=\#\{\gamma\in\Gamma\,:\,d(x,\gamma y)\}$ for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group $\Gamma$ of motions in X, such that $\Gamma\backslash X$ has finite volume. We show that \[ \displaystyle N(x,y;r) = \sum_{j=0}^m c_j \varphi_j(x) \varphi_j(y) e^{(\rho+\nu_j)r} + O_{x,y,\varepsilon}\left( e^{(2\rho n/(n+1) +\varepsilon)r}\right) \] as $r\rightarrow\infty$, for each $\varepsilon>0$. The constant $2\rho$ corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions $\varphi_j\in L^2(\Gamma\backslash X)$ of the Laplacian, such that the eigenvalues $\rho^2-\nu_j^2$ are less than $4n\rho^2/(n+1)^2$.