The resolvent trace formula for rank one Lie groups (Q1811392)

Let \(G\) be a connected semisimple Lie group of real rank one with finite center and \(K\) be its maximal compact subgroup. Then \(X = G/K\) is a Riemannian symmetric space of non-compact type. Introduce, following \textit{R. J. Miatello} and \textit{N. R. Wallach} [J. Differ. Geom. 36, 663-698 (1992; Zbl 0766.53044)], a family of bi-\(K\)-invariant functions \(Q_s\), \(s \in C\), on \(G\) and define then a function \(Q_{r,s}\) on \(G\) via the \(r\)-fold convolution of \(Q_s\). A formula of \(Q_{r,s}\) in terms of a derivative of the hypergeometric series is derived and the integral of the Poincaré series is computed via partitioning it into the local contributions for \(\Gamma\)-conjugacy classes. Consequently, the integral is evaluated by means of the logarithmic derivative of the Selberg zeta function for \(\Gamma \setminus X\) arriving thus at the resolvent trace formula.