Fourier Coefficients of Automorphic Forms and Integrable Discrete Series

TL;DR

This paper investigates Fourier coefficients of Poincaré series linked to integrable discrete series of semisimple groups over Q, constructing explicit automorphic realizations with non-zero Fourier coefficients, and relating results to prior work in special cases.

Contribution

It provides explicit constructions of automorphic cuspidal realizations for integrable discrete series using Fourier coefficients of Poincaré series, extending previous results to broader classes of groups.

Findings

Constructed automorphic realizations with non-zero Fourier coefficients.

Connected results to Li's work for Sp_{2n}(R).

Extended Poincaré series analysis to quasi-split groups over Q.

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].