In this paper we characterize irreducible generic representations of SO 2n+1 (k) (where k is a p-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of SO 2n+1 (A) (where A is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem); and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of SO 2n+1 (k).

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.

In memory of Steve RallisWe study the location of possible poles of a family of residual Eisenstein series on classical groups. Special types of residues of those Eisenstein series were used as key ingredients in the automorphic descent constructions of Ginzburg, Rallis and Soudry and in the refined constructions of Ginzburg, Jiang and Soudry. We study the conditions for the existence of other possible poles of those Eisenstein series and determine the possible Arthur parameters for the residual representations if they exist. Further properties of those residual representations and their applications to automorphic constructions will be considered in our future work.

Abstract. Fourier coefficients of automorphic representations π of Sp 2n (A) are attached to unipotent adjoint orbits in Sp 2n (F ), where F is a number field and A is the ring of adeles of F . We prove that for a given π, all maximal unipotent orbits that gives nonzero Fourier coefficients of π are special, and prove, under a well acceptable assumption, that if π is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is F -anisotropic as algebraic group over F . These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudry on the subject. As a consequence, we obtain constraints on those maximal unipotent orbits if F is totally imaginary, further applications of which to the discrete spectrum with the Arthur classification will be considered in our future work.

Abstract. We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.