6

INTRODUCTION

Notice that here we have n!=7i2] instead of ftW i n (7)-

In order to prove meromorphic continuation of (7), we have to specify the analytic

nature of f at archimedean primes. In this book we treat, for simplicity, only the

case where F is totally real, though the general case can be handled by the same

ideas. First, the archimedean localization of G+(V) acts on a symmetric space

which we denote by Z^. Then we assume that f corresponds to functions on Z^

that are eigenfunctions of all invariant differential operators. The space of such

eigenfunctions contains a spherical function, parametrized by finitely many complex

numbers {avi}. Then our main theorem says:

(G) There exists a product G(s) of gamma functions, which can be explicitly

determined by the parameters {aVi} and \- such that Q(s)Z(s, f, x) can De con-

tinued as a meromorphic function to the whole s-plane with finitely many poles;

moreover, Q(s)Z(s, f, \) JS entire if ipx2 ^ 1.

If ip is trivial, then f is a function on G+(V)A/F£, SO that it is essentially a

function g on SO^, and Z(s, g, x) — Z(s, f, \ ) . Therefore our result for Z(s, f, \ )

includes that for Z(s, g, x)- However, we state the results separately as Theorems

17.15 and 22.7 for the reason which will be explained below.

We also consider an Eisenstein series on G+(W) — GJr{W, uS) with a space W

containing V such that dim(W) = dim(V) + 2m and with uo given by

0 0 2 - x l

r

0 ip 0

2 ~ 1 l

m

0 0

Then O" has a natural parabolic subgroup P" consisting of the elements whose

lower left three blocks under the diagonal blocks are 0. Then we define a parabolic

subgroup V in G+(W) by V = {a e G+(W) \ r{a) G Pu}. Taking a subgroup Ju

of GJ*~(W)A similar to J, we can define, under a natural condition of consistency of

J and J", an Eisenstein series E(x, s; f, x, J") for (x, s) G G+(W)A x C. This is

similar to the series studied in [S97] and [S00] in the symplectic and unitary cases.

Then we shall prove (Theorem 24.7):

(H) The product

[(m+n)/2]-l

Z(2s, f,

X

)E(x, s; f,

x

, J") U Lc(4s - 2j, ^x2)

j = [n/2]

times an explicitly dehned product of gamma functions can be continued to a

meromorphic function on the whole s-plane with finitely many poles, where f'(x) =

4,(v(x))f(x).

If ip = 1, then the above Eisenstein series can be viewed as an Eisenstein series

on SO". However, we first treat Euler products and Eisenstein series on O^ and

Ou (or on SO^ and SO"), and then those on G+(V) and G+{W) after that.

We believe that this approach, if redundant to some extent, makes our exposition

easier to read. In fact, our proof of meromorphic continuation requires pullbacks

of Eisenstein series on a split orthogonal or Clifford group. The idea is similar to

what was done in [S97] and [S00], and each pullback can be analyzed more or less