New vectors for GSp(4): a conjecture and some evidence

In this paper we present and state evidence for a conjecture on the existence and properties of new vectors for generic irreducible admissible representations of GSp(4, F ) with trivial central character for F a nonarchimedean field of characteristic zero. To summarize the conjecture, let O be the ring of integers of F and let P be the prime ideal of O. We define, by a simple formula, a sequence of compact open subgroups K(Pn) of GSp(4, F ) indexed by nonnegative integers n. The first group K(O) is GSp(4,O). The second group K(P) is the other maximal compact subgroup of GSp(4, F ), up to conjugacy, and is called the paramodular group. Automorphic forms for the global version of this group have been considered by T. Ibukiyama and his collaborators in a number of papers dealing with a genus two version of Eichler’s correspondence and old and new forms. In general, we refer to K(Pn) as the paramodular group of level Pn. Given a generic irreducible admissible representation π of GSp(4, F ) with trivial central character, we consider the space of vectors fixed by each K(Pn). The conjecture for π makes three assertions. First, for some nonnegative n, the space of K(Pn) fixed vectors is nonzero; second, if Nπ is the smallest such n, then the space of K(PNπ) fixed vectors is one dimensional; and third, this one dimensional space contains a vector Wπ whose Novodvorsky zeta integral gives the Novodvorsky L-factor of the representation: Z(s,Wπ) = L(s, π). We call Wπ the new vector of π. Zeta integrals depend on a choice of Whittaker model, which depends on a choice of nondegenerate character: we make a choice independent of π. Evidently, the conjecture is similar to the theory of new vectors for generic irreducible admissible representations of GL(2, F ) with trivial central character. Just as for GL(2, F ), there is a simple relation between new vectors and