Explicit Formulas for the Waldspurger and Bessel Models

Daniel Bump, Solomon Friedberg, and Masaaki Furusawa 

MSRI #1995-002

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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 \cite{Wa1,Wa2}, and arose in his profound studies of
the Shimura correspondence.  If $n=2$, they were considered by
Novodvorsky and Piatetski-Shapiro \cite{NP}, who called them {\it
Bessel models}, and for general $n$ they were studied by Novodvorsky
\cite{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~\cite{An}, Furusawa~\cite{Fu1}, and
Sugano \cite{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~\cite{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 $\pi$ on $GL(n)$, and we
show that its Bessel period (6.2) is essentially 
a product of L-series
$$L\big(n(s-1/2)+1/2,\pi\big)\,L\big(n(s-1/2)+1/2,\pi\otimes\eta\big),$$
where $\eta$ is a quadratic character.
This result generalizes work for $n=2$ and base field $\bold Q$ 
of Mizumoto~\cite{Mi} and B\"ocherer~\cite{B\"o}, and puts it in
a representation-theoretic context.

This global application is closely
related to the results of Bump, Friedberg, and Hoffstein~\cite{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 \cite{BFH2} has the following consequence: 
if one forms the metaplectic Eisenstein series on the double
cover of $GSp(2n)$ with a cuspidal automorphic representation
$\pi$ of $GL(n)$ (which is possible because the cover splits over
$GL(n)\subset Sp(2n)$), the Whittaker coefficients of this Eisenstein
series are quadratic twists of the standard L-function of $\pi$. The
close relation between these two computations is a reflection of the 
following result
of Furusawa~\cite{Fu2}, generalizing the case $n=2$ in
Piatetski-Shapiro and Soudry~\cite{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 \cite{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 $GSp(6)$ (\cite{BG})
and on $SO(7)$ (\cite{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~\cite{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~\cite{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.
 
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