On the genericity of cuspidal automorphic forms of SO_2n+1

Abstract

We study the irreducible generic cuspidal support up to near equivalence for certain cuspidal automorphic forms of SO_2n+1 (Theorem 3.2 and Theorem 4.1), by establishing refined arguments in the theory of local and global Howe duality and theta correspondences ([Jiang, D., Soudry, D., The local converse theorem for SO(2n + 1) and applications, Ann. Math. (2) 157 (2003), no. 3, 743–806.], [Furusawa, M., On the theta lift from , J. reine angew. Math. 466 (1995), 87–110.]) and in the theory of Langlands functoriality ([Cogdell, J., Kim, H., Piatetski-Shapiro, I., Shahidi, F., On lifting from classical groups to GL(n), IHES Publ. Math. 93 (2001), 5–30.], [Jiang, D., Soudry, D., The local converse theorem for SO(2n + 1) and applications, Ann. Math. (2) 157 (2003), no. 3, 743–806.], [Ginzburg, D., Rallis, S., Soudry, D., Generic automorphic forms on SO(2n + 1): functorial lift to GL(2n), endoscopy, and base change, Internat. Math. Res. Notices 14 (2001), 729–764.]). The results support a global analogy and generalization of a conjecture of Shahidi on the genericity of tempered local L-packets (Conjecture 1.1). The methods are expected to work for other classical groups.