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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.

Let G be a connected reductive complex algebraic group, and E a complex elliptic curve. Let GE denote the connected component of the trivial bundle in the stack of semistable Gbundles on E. We introduce a complex analytic uniformization of G_E by adjoint quotients of reductive subgroups of the loop group of G. This can be viewed as a nonabelian version of the classical complex analytic uniformization E=C^∗/q^Z. We similarly construct a complex analytic uniformization of G itself via the exponential map, providing a nonabelian version of the standard isomorphism C^∗= C/Z, and a complex analytic uniformization of G_E generalizing the standard presentation E = C/(Z ⊕ Zτ). Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from Sh_N(G_E) (the semistable part of the automorphic category) to IndCoh_{Nˇ}(Locsys_{Gˇ}(E)) (the spectral category).

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