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 $\ne 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=S{p}_{2n}(\mathbb{R})$, 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].