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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.

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The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions

In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of (generalized) affine Deligne-Lusztig varieties $X(\mu, b)_J$ for a $p$-adic group $G$ and its parahoric subgroup $P_J$. We show that $X(\mu, b)_J \neq \emptyset$ if and only if the group-theoretic version of Mazur's inequality is satisfied. In the process, we obtain a generalization of Grothendieck's conjecture on the closure relation of $\s$-conjugacy classes of a twisted loop group.