We study the irreducible generic cuspidal support up to near equivalence for certain cuspidal automorphic forms of SO2n+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.
Bibliographical noteFunding Information:
The first named author is supported in part by the NSF grant DMS-0400414, and by the Distinguished Visiting Professorship at The Institute of Mathematics, The Chinese Academy of Science. The second named author’s research is supported by the Israel Science Foundation. We would like to thank the referee for very helpful remarks.