Baiying Liu

Genericity of Representations of p-Adic

Let G be the F-rational points of the symplectic group Sp2n, where F is a non-Archimedean local field of characteristic 0. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Langlands functorial lifting from irreducible generic representations of G to irreducible representations of GL2n+1(F). Jiang and Soudry constructed the descent map from irreducible supercuspidal representations of GL2n+1(F) to those of G, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying SO2n+1 as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter φ ∈ Φ(G), we construct a representation σ such that φ and σ have the same twisted local factors. As one application, we prove the G-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter φ ∈ Φ(G) is generic, i.e., the representation attached to φ is generic, if and only if the adjoint Lfunction of φ is holomorphic at s = 1. As another application, we prove for each Arthur parameter ψ, and the corresponding local Langlands parameter φψ , the representation attached to φψ is generic if and only if φψ is tempered.

Sp_{2n}

In the theory of automorphic descents developed by Ginzburg, Rallis and Soudry in [GRS11], the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays important roles. Started from [JLZ13], the authors are looking for more general residual representations, which may yield more general theory of automorphic descents. In this paper, we investigate the structure of Fourier coefficients of certain residual representations of symplectic groups, corresponding to certain interesting families of global Arthur parameters. On one hand, the results partially confirm a conjecture proposed by the first named author in [J14] on relations between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. On the other hand, the results of this paper can be regarded as a first step towards more general automorphic descents for symplectic groups, which will be considered in our future work.

and Local Langlands Parameters

Abstract In this paper, we reprove a global converse theorem of Cogdell and Piatetski-Shapiro using purely global methods.