MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS 11

(1) First we have a functional equation under s — 1 — 5, obtained from (3.3).

Because of the power of \d\ that is introduced we have, upon substituting

into (3.1), w - w + r(s - §). Thus

(3.5) Z(s,w) - Z ( l - s , w + r ( s - l / 2 ) ) .

(Strictly speaking we should write the right hand side as Z(\ — s, w-\-r(s —

1/2)) as 7r is replaced by its contragredient.)

(2) We also have a functional equation under the transformation w — » 1 — w,

obtained from (3.4). Applying this to (3.2) yields a transformation

(3.6) Z(s,w) - Z(s + w- 1/2,1 -w).

Note that each of these functional equations goes hand in hand with an exten-

sion of Z(s, w), originally defined by an absolutely convergent series in Sft(s), $l(w)

1, to a larger region. It is convenient to think of these transformations as operating

(repeatedly) on a region of definition to extend the function to a larger region, and

we will do so below, but strictly speaking one obtains first the continuation to the

larger region (by Phragmen-Lindelof), and then the functional equation on this

larger region.

Writing these functional equations carefully would require writing the archime-

dean factors and also describing a suitable scattering matrix; for the heuristic this

level of detail is not needed.

One can apply the functional equations (3.5) and (3.6) successively. They

generate a finite group of functional equations for GL(1), GL(2) and GL(3), i.e for

r = 1, 2, 3 but an infinite group for GL(4) (in fact an affine Weyl group) and higher.

This suggests that it should be possible to define a precise, non-heuristic, Z(s,w)

that continues to

C2

for GL(1), GL(2) and GL(3) but that significant obstructions

may appear for GL(4) and higher.

To go farther, let us consider poles. We expect that there is a pole at w = 1,

since

C(w)

arises in equation (3.2) when d = 1. This polar line is reflected by the

functional equations into a collection of polar lines that will be finite if r = 1, 2, 3 and

infinite if r 4 (see [16]). For any fixed SQ the possibility of a pole at s = so,

w

— 1

can be investigated by computing the sum of the contributions from the polar lines

that intersect (so, 1). If Z(s,w) does in fact have a pole at (50,1), then, by (3.1),

this implies the non-vanishing of L(SQ,7T X Xd) for infinitely many Xd- Similarly if

one can continue to (s,w) = (1/2,1) and if all epsilon factors at 1/2 are —1 then

one can differentiate with respect to s and set s = 1/2. There should still be a pole

at w = 1 provided that the different polar divisors do not cancel when s = 1/2. In

that case, one may then obtain a non-vanishing theorem for 1/(1/2, TT X Xd) from the

pole of -^Z(s,w) at s = 1/2, w — 1. Standard methods involving contour integrals

can also give mean value theorems.

In the case of GL(4) and higher the group of functional equations is infinite.

If we take this infinite group and use it to translate the line w = 1, the poles

accumulate and create a barrier to continuation. See [16], Section 4, for some

elaboration of this point. Because of this we do not expect continuation to all of

C2

when r 4. However, if we could get continuation up to the conjectured barrier,

that would be very significant; we would get a tremendous amount of information

(Lindelof in twisted aspect, simultaneous non-vanishing at the center of the critical

strip). At the moment this problem seems extremely challenging.