Spinor L-functions, theta correspondence, and Bessel coefficients

Abstract

In this paper we prove two seemingly unrelated theorems. First we establish the entireness of the spinor L-functions of certain automorphic cuspidal representations of the similitude symplectic group of order four over the rational numbers. We also prove a theorem related to the existence of Bessel models for generic discrete series representations of the same group over the real numbers. The two results are linked by the method of proof; in both cases it is based on the pull-back of an appropriately chosen global Bessel functional via the theta correspondence for the dual pair (GO(2, 2),GSp(4)). The first main theorem is related to analytic properties of spinor L-functions. We prove the entireness of the spinor L-function for those generic automorphic cuspidal representation which satisfy a condition at the archimedean place (see below). Our study of the spinor L-function is based on an integral representation which works for generic representations. These integrals which were introduced by M. Novodvorsky in the Corvallis conference [26] serve as one of the few available integral representations for the Spinor L-function of GSp(4). Some of the details missing in Novodvorsky’s original paper have been reproduced in Daniel Bump’s survey article [4]. Further details have been supplied by [40]. Novodvorsky’s integral was first generalized by Ginzburg [10], and further generalized by Soudry [39], to orthogonal groups of arbitrary odd degree. In light of the results of [40], it is sufficient to study the integral of Novodvorsky at the archimedean place. Archimedean computations are often forbidding, and unless one expects major simplifications due to the nature of the parameters, the resulting integrals are often quite hard to manage. In our case of interest, the work of Moriyama [25] benefits from exactly such simplifications when he treats the case of cuspidal representations with archimedean components in the generic (limit of) discrete series. In this work, we concentrate on those archimedean representations for which direct computations have yielded very little. For this reason, our methods are a bit indirect, in fact somewhat more indirect than what at first seems necessary. Our method is based on the theta correspondence. First we observe in Lemma 2.2 that Novodvorsky’s integral is in fact a split Bessel functional. Then in 2.1 we pull the Bessel functional back via the theta correspondence for the dual reductive pair (GO(2, 2),GSp(4)), and prove that the resulting functional on GO(2, 2) is Eulerian. On the other hand,