On the genericity of cuspidal automorphic forms of $\mathbf{SO}\bm{(2n+1)}$, II

Abstract

This paper is a continuation of our previous work (D. Jiang and D. Soudry, On the genericity of cuspidal automorphic forms on${\rm SO}_{2n+1}$, 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 $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models. We also give a characterization of the vanishing of the central value of the standard $L$-function of $\mathrm{SO}_{2n+1}(\mathbb{A})$ in terms of theta correspondence. As a result, we obtain the weak Langlands functorial transfer from $\mathrm{SO}_{2n+1}(\mathbb{A})$ to $\mathrm{GL}_{2n}(\mathbb{A})$ for irreducible cuspidal automorphic representations of $\mathrm{SO}_{2n+1}(\mathbb{A})$ with special Bessel models.