On the genericity of cuspidal automorphic forms of SO(2n+1), II

This paper is a continuation of our previous work (D. Jiang and D. Soudry, On the genericity of cuspidal automorphic forms on SO_2n+1, J. reine angew. Math., to appear). We extend Moeglin's results (C. Moeglin, J. Lie Theory 7 (1997), 201-229, 231-238) from the even orthogonal groups to old orthogonal groups and complete our proof of the CAP conjecture for irreducible cuspidal automorphic representations of SO_2n+1(double-struck A sign) with special Bessel models. We also give a characterization of the vanishing of the central value of the standard L-function of SO_2n+1 (double-struck A sign) in terms of theta correspondence. As a result, we obtain the weak Langlands functorial transfer from SO_2n+1(double-struck A sign) to GL_2n(double-struck A sign) for irreducible cuspidal automorphic representations of SO_2n+1(double-struck A sign) with special Bessel models.