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 = ⊗vσv be an irreducible tempered cuspidal automorphic representation of O(X,A). We give a sufficient condition for the nonvanishing of the theta lift Θn(σ) of σ to the symplectic group Sp(n,A) (2n by 2n matrices) for 2n ≥ dimX for a large class of X. As a corollary, we show that if 2n = dimX and all the local theta lifts Θn(σv) are nonzero, then Θn(σ) is nonzero if the standard L-function LS(s, σ) is nonzero at 1, and Θn−1(σ) is nonzero if LS(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. Let H and G be reductive linear algebraic groups defined over a number field F with ring of adeles A. A fundamental problem in the theory of automorphic forms is to investigate the existence of liftings of irreducible automorphic representations of H(A) to irreducible automorphic representations of G(A). Here, by a lifting we mean a map from a subset of the set of irreducible automorphic representations of H(A) to the set of irreducible automorphic representations of G(A) such that if σ maps to π, then the local unramified components of σ and π are related by the functorality principle of Langlands, or by a natural extension of this principle in the nonconnected case. The existences of such liftings have important consequences in number theory. Several programs exist for constructing and investigating liftings, and in this paper we shall be concerned with one such method, the theta correspondence. The theta correspondence provides liftings in the case that H and G form a reductive dual pair. While the theta correspondence thus applies only to a limited number of pairs H and G, liftings constructed via the theta correspondence are part of a rich structure with roots in the classical theory of automorphic forms. Examples of structures related to theta lifts include the Siegel-Weil formula and its consequences, period integrals, Fourier coefficients, and the behavior of L-functions at special points. In this work we consider the case when H is the orthogonal group of an even dimensional quadratic space and G is a symplectic group. If 1991 Mathematics Subject Classification. Primary 11F27; Secondary 11F70, 11F46. This research was partially supported by NSERC research grant OGP0183677. 1