On the residues and special values of Poincaré series (Q1279767)

Let \(G\) be the universal cover of SL\(_2(\mathbb{R})\), \(\Gamma\) a lattice in \(G\) and \(\chi\) a unitary character of \(\Gamma\). The aim of the work under review is the construction of square integrable \((\Gamma,\chi)\)-automorphic forms on \(G\). To that end the authors introduce a meromorphic family of functions defined by means of a matrix entry of a principal series representation. This family is first defined in a half-plane and then extended to the whole complex plane by meromorphic continuation. (Classically this family is known as a family of Poincaré series and its meromorphic continuation is a crucial part of the classical theory.) The main issue of the present paper is the proof of various completeness results in \(L^2_d(\Gamma\setminus G,\chi)\) for the residues and special values of the family under consideration. This implies in particular completeness results for holomorphic (or antiholomorphic) forms of small weight.