EXPLICIT FORMULAS FOR THE WALDSPURGER AND BESSEL MODELS

Daniel Bump, Solomon Friedberg, and Masaaki Furusawa

Typeset by AMS-TEX Abstract. In this paper we will study certain models of irreducible admissible representations of the split special orthogonal group SO(2n + 1) over a nonarchimedean local field. If n = 1, these models were considered by Waldspurger [Wa1,Wa2], and arose in his profound studies of the Shimura correspondence. If n = 2, they were considered by Novodvorsky and Piatetski-Shapiro [NP], who called them Bessel models, and for general n they were studied by Novodvorsky [No]. In the works cited, these authors established the uniqueness of these models; in this paper we establish functional equations and explicit formulas for them. In general, these models arise from a variety of Rankin-Selberg integrals (for example, those of Andrianov [An], Furusawa [Fu1], and Sugano [Su]), and the results of this paper will naturally have applications to the study of L-functions. Moreover, these models arise in the study of the theta correspondence between SO(2n+1) and the double cover of Sp(2n), and they will therefore be of importance in generalizing the work of Waldspurger (see Furusawa [Fu2]).

In the final Section, we present a global application of the explicit formulas: we consider the Eisenstein series (6.1) on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n), and we show that its Bessel period (6.2) is essentially a product of L-series

L(n(s − 1/2) + 1/2, π) L(n(s − 1/2) + 1/2, π ⊗ η),

where η is a quadratic character. This result generalizes work for n = 2 and base field Q of Mizumoto [Mi] and Böcherer [Bö], and puts it in a representation-theoretic context.

This global application is closely related to the results of Bump, Friedberg, and Hoffstein [BFH2]. That paper computes the spherical Whittaker functions on the metaplectic double cover of Sp(2n). (Whittaker models on that group are also unique.) The computation in [BFH2] has the following consequence: if one forms the metaplectic Eisenstein series on the double cover of GSp(2n) with a cuspidal automorphic representation π of GL(n) (which is possible because the cover splits over GL(n) ⊂ Sp(2n)), the Whittaker coefficients of this Eisenstein series are quadratic twists of the standard L-function of π. The close relation between these two computations is a reflection of the following result of Furusawa [Fu2], generalizing the case n = 2 in Piatetski-Shapiro and Soudry [PS]: the (special) Bessel coefficient of a cusp form on SO(2n + 1) essentially agrees with the Whittaker coefficient of the theta lift on the double cover of Sp(2n). If instead of a cusp form one considers the Eisenstein series (6.1), the theta correspondent on the metaplectic group is the metaplectic Eisenstein series, and our calculation implies that this result of Furusawa for cusp forms is true for these Eisenstein series also. (Our calculation of the Bessel period is in fact direct and independent of [Fu2].)

These results should have an application to the nonvanishing of L-functions under quadratic twists. Namely, there are Rankin-Selberg integrals on the double cover of GSp(6) ([BG]) and on SO(7) ([Gi]) unfolding to Dirichlet series involving the Whittaker (resp. Bessel) periods described above, that is, to Dirichlet series whose individual coefficients are the quadratic twists of a standard GL(3) L-series. (The two constructions give Dirichlet series whose individual coefficients are Euler products which agree at almost all places.) Arguing as in [BFH1], one should be able to show that an infinite number of these quadratic twists are nonzero. In fact, these integrals are the next members of a series beginning with Siegel’s calculation of the Mellin transform of a metaplectic GL(2) Eisenstein series and including integrals of Hecke type on the double cover of GSp(4) (due in a nonmetaplectic context to Novodvorsky; see [BFH1]) and on SO(5) (due to Maass). The elucidation of this scenario owes much to discussions with Duke, Ginzburg, Goldfeld, and Hoffstein. In particular, the verification that our results could be applied to the evaluation of (6.2) was first worked out in conversation with Ginzburg.