Nonvanishing of Global Theta Lifts from Orthogonal Groups

Let X be an even dimensional symmetric bilinear space defined over a totally real number field F with adeles A, and let σ = ®_vσ_v be an irreducible cuspidal automorphic representation of 0(X, A) tempered at the finite places. We give a sufficient condition for the nonvanishing of the theta lift O_n(σ) of σ to the symplectic group Sp(n, A) (2n by 2n matrices) for 2n ≥ dim X for a large class of X. As a corollary, we show that if 2n = dim X and all the local theta lifts O_n(σ_v) are nonzero, then ©_n(σ) is nonzero if the standard L-function L ^S(s, σ) is nonzero at 1, and ☉_{n_1}(σ) is nonzero if L^S(s, σ) has a pole at 1. The proof uses only essential structural features of the theta correspondence, along with a new result in the theory of doubling zeta integrals.