Fourier Coefficients of Automorphic Forms and Integrable Discrete Series

Abstract

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=Sp_{2n}(\mathbb R)$, we relate our work to that of Li [15]. For $\mathcal G$ quasi--split over $\mathbb Q$, we relate our work to the result about Poincar\' e series due to Khare, Larsen, and Savin [16].