Resolvent and lattice points on symmetric spaces of strictly negative curvature (Q1962561)

Let \(G\) be a semisimple Lie group of rank one over \(\mathbb{R}\), \(K\) a maximal compact subgroup of \(G\) and \(X= G/K\) the associated Riemannian symmetric space of strictly negative curvature. Moreover, let \(\Gamma\) be a discrete subgroup of \(G\) of finite volume. The authors study the asymptotic behaviour of the lattice point counting function \[ N(x,y;r):= \#\{\gamma\in \Gamma\mid d(x,\gamma y)< r\} \] as \(r\to \infty\), for fixed \(x,y\in X\). The main result of the work under review links the asymptotic behaviour of \(N(x,y,r)\) with spectral data of the Laplacian on \(L^2(\Gamma \setminus X)\). More precisly, \[ N(x,y;r)= c_0 e^{2\rho r}+ \sum_{j=1}^m c_j \varphi_j(x) \varphi_j(y) e^{(\rho+ \nu_j)r}+ O\left( \exp \Biggl( \biggl( \frac{2\rho n}{n+1}+ \varepsilon\biggr) r\Biggr)\right) \] as \(r\to \infty\) for each \(\varepsilon> 0\). Here, the constant \(2\rho\) corresponds to the sum of the positive roots of \(G\) and \(n= \dim X\). The sum extends over an orthonormal system of real-valued eigenfunctions \(\varphi_j\in L^2 (\Gamma \setminus X)\) of the Laplacian such that the associated ``exceptional'' eigenvalues \(\rho^2- \nu_j^2\) belong to the interval \(]0,\rho^2[\). The constant \(c_0\) is determined by the volume of \(\Gamma \setminus X\), and \(c_j\) involves the value \(c(\nu_j)\) where \(c(\cdot)\) is the Harish-Chandra \(c\)-function \((j= 1,\dots, m)\).