Genericity of Representations of p-Adic Sp2n and Local Langlands Parameters

Liu, Baiying

Canadian journal of mathematics, Tome 63 (2011) no. 5, pp. 1107-1136

Résumé

Let $G$ be the $F$ -rational points of the symplectic group $S{{p}_{2n}}$ , 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 $G{{L}_{2n+1}}\left( F \right)$ . Jiang and Soudry constructed the descent map from irreducible supercuspidal representations of $G{{L}_{2n+1}}\left( F \right)$ 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 $S{{O}_{2n+1}}$ as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter $\phi \,\in \,\Phi \left( G \right)$ , we construct a representation $\sigma $ such that $\phi $ and $\sigma $ 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 $\phi \,\in \,\Phi \left( G \right)$ is generic, i.e., the representation attached to $\phi $ is generic, if and only if the adjoint $L$ -function of $\phi $ is holomorphic at $s\,=\,1$ . As another application, we prove for each Arthur parameter $\psi $ , and the corresponding local Langlands parameter ${{\phi }_{\psi }}$ , the representation attached to ${{\phi }_{\psi }}$ is generic if and only if ${{\phi }_{\psi }}$ is tempered.

Détail

DOI : 10.4153/CJM-2011-017-2

Mots-clés : 22E50, 11S37, generic representations, local Langlands parameters