Dihua Jiang 1,
1School of Mathematics, University of Minnesota, Minneapolis, MN55455, USA.
dh.jiang@malk.umn.edu David Soudry 22School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978 Israel.
Abstract
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.
