The resolvent trace formula for rank one Lie groups
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.
Lie group of rank one spherical functions
scientific article; zbMATH DE number 1925758