Fourier coefficients of automorphic forms and integrable discrete series

Let G   be the group of RR-points of a semisimple algebraic group GG defined over QQ. Assume that G   is connected and noncompact. We study Fourier coefficients of Poincaré series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients ≠0, of integrable discrete series in families of congruence subgroups. In the case of G=Sp2n(R)G=Sp2n(R), we relate our work to that of Li [14]. For GG quasi-split over QQ, we relate our work to the result about Poincaré series due to Khare, Larsen, and Savin [12].