The existence of the well-known Jacquet–Langlands correspondence was established by
Jacquet and Langlands via the trace formula method in 1970. An explicit construction
of such a correspondence was obtained by Shimizu via theta series in 1972. In
this paper, we extend the automorphic descent method of Ginzburg–Rallis–Soudry
to a new setting. As a consequence, we recover the classical Jacquet–Langlands correspondence
for PGL(2) via a new explicit construction.
The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.