HIRONORI SAKUNO (AFTER D.BUMP)
Abstract. The main topics which we shall consider are the theory of Whittaker functions, their differential equations, and their analytic continuation and func- tional equations ; Fourier expansions on GL(3) ; the Fourier expansionsof the Eisenstein series, and the theory of Ramanujan sums on GL(3), which arise in the Fourier expansions ; the analytic continuation and functional equations of the Eisenstein series ; the polar divisorof the Eisenstein series ; the interpretation of the Fourier coefficients of the Eisenstein series as generalized divisor sums, expressed in terms of Schur polynomials.
1. Whittaker functions on GL(3,R)
We will exhibit six linearly independent solutions of the Whittaker differential equations as generalized hypergeometric series, and one solution which satisfies the growth condition, the latter as either a Mellin–Barnes integral, or as a linear combi- nation of the six hypergeometric Whittaker functions.
Let G = GL(3,R), K = O(3) be the maximal compact subgroup of orthogonal matrices, and let Z be the center of G. We introduce coordinates on H=G/ZK as follows
τ =
y1y2 y1x2 x3 y1 x1 1
, where y1, y2 >0.
The Weyl group W of G is the group of the six matrices : w0 =
1 1
1
, w1 =
−1
−1
−1
, w2 =
−1
−1
−1
w3 =
−1
−1
−1
, w4 =
1 1 1
, w5 =
1 1
1
.
Let us consider the function :
I(τ) =I(ν1,ν2)(τ) = y12ν1+ν2y2ν1+2ν2
1
on H. The action ofW on the parameter ν1, ν2 is defined by requiring I(ν1−1
3,ν2−13)(τ) = I(µ1−1
3,µ2−13)(wτ), (x1 =x2 =x3 = 0), when (µ1, µ2) = w·(ν1, ν2). Thus we have
w0·(ν1, ν2) = (ν1, ν2), w1·(ν1, ν2) = (2
3−ν2,2
3−ν1), w2·(ν1, ν2) = (ν1+ν2− 1
3,2
3 −ν2), w3·(ν1, ν2) = (2
3−ν1, ν1+ν2− 1 3), w4·(ν1, ν2) = (1−ν1−ν2, ν1), w5·(ν1, ν2) = (ν2,1−ν1−ν2).
We shall consiser the following G–invariant differential operators on H
∆1 =y12 ∂2
∂y21 +y22 ∂2
∂y22 −y1y2 ∂2
∂y1∂y2 +y12(x22+y22) ∂2
∂x23 +y21 ∂2
∂x21 +y22 ∂2
∂x22 + 2y12x2 ∂2
∂x1∂x3,
∆2 =−y12y2 ∂3
∂y12∂y2
+y1y22 ∂3
∂y1∂y22 −y13y22 ∂3
∂x23∂y1
+y1y22 ∂3
∂x22∂y1 −2y12y2x2 ∂3
∂x1∂x3∂y2 + (−x22+y22)y12y2 ∂3
∂x23∂y2
−y21y2 ∂3
∂x21∂y2 + 2y12y22 ∂3
∂x1∂x2∂x3 + 2y12y2x2 ∂3
∂x2∂x23 +y12 ∂2
∂y21 −y22 ∂2
∂y22 + 2y21x2
∂2
∂x1∂x3 + (x22+y22)y12 ∂2
∂x23 +y12 ∂2
∂x21 −y22 ∂2
∂x22. Then we have
∆1I =λI, ∆2I =νI,
where λ = −1−(αβ +βγ +γα), ν = −αβγ, α = −ν1 −2ν2 + 1, β = −ν1 +ν2, γ = 2ν1+ν2−1. Note that the action of the Weyl group permutesα, β, γ.
Definition 1.1. The Whittaker function is a function F on H such that
(1) F is an eigenfunction of ∆1 and ∆2 with the same eigenvalues as I, that is :
∆1F =λF, ∆2F =µF, (2) We have
F
1 x2 x3 1 x1 1
τ
=e(x1+x2)F(τ),
where we set e(x) = exp(2πıx).
We set
F(y1, y2) =F
y1y2 y1
1
.
Then the conditions (1) and (2) imply for F(y1, y2) the differential equations :
{
y12 ∂2
∂y12 +y22 ∂2
∂y22 −y1y2 ∂2
∂y1∂y2 −4π(y12+y22)
}
F(y1, y2) = λF(y1, y2),
{
−y12y2 ∂3
∂y12∂y2
+y1y22 ∂3
∂y1∂y22 + 4π2y12y2 ∂
∂y2 −4π2y1y22 ∂
∂y1
+y12 ∂2
∂y21 −y22 ∂2
∂y22 −4π2y12+ 4π2y22
}
F(y1, y2) = µF(y1, y2).
Note that these equations are invariant under the action of the Weyl group, since the parameters λ and µ are symmetric polynomials in α, β, γ.
Theorem 1.1. (1) The dimension of the space of Whittaker functions on H is six.
(2) We may construct six linearly independent solutions of these differential equa- tions, as generalized hypergeometric series :
M(ν1,ν2)(y1, y2) =
∑∞ n1,n2=0
(3ν1+ 3ν2 2
)(n1+n2)−(n1)−(n2)
(πy1)2n1(πy2)2n2 n1!n2!
(3ν1+ 1 2
)(n1)(
3ν2+ 1 2
)(n2) ,
unless ν1 = 1
3, ν2 = 1
3, or 1−ν1 −ν2 = 1
3. Here we use the notations a(n) =
∏n−1
k=0(a+k).
(3)There exist the moderate growth Whittaker function Wν1,ν2 : W(ν1,ν2)(y1, y2) = 1
4(2πı)2
∫ σ+ı∞
σ−ı∞
∫ σ+ı∞
σ−ı∞ V(s1, s2)(πy1)1−s1(πy2)1−s2ds1ds2, where the line of integration are taken to the right of all poles of the integrand, and
V(s1, s2) = Γ(s12+α)Γ(s1+β2 )Γ(s1+γ2 )Γ(s2−2α)Γ(s2−2β)Γ(s2−2γ)
Γ(s1+s2 2) .
2. Fourier expansion of automorphic forms
We show that an automorphic form has a Fourier expansion, the coefficients form- ing a two–dimensional array.
Definition 2.1. Let Γ =GL(3,Z). For ν1, ν2 ∈C, an automorphic form of type ν1, ν2 is a function ϕ on H such that
(1) ϕ(γτ) = ϕ(τ), for all γ ∈Γ,τ ∈ H, (2) ∆1ϕ=λϕ, ∆2ϕ=µϕ,
(3)There exist constants n1, n2 such that ϕ
y1y2 y1
1
y1n1y2n2
is bounded on {(y1, y2)∈R×R|y1, y2 >1}. If furthermore
∫ 1
0
∫ 1
0
ϕ
1 ξ3 1 ξ1 1
τ
dξ1dξ3 = 0,
∫ 1
0
∫ 1
0
ϕ
1 ξ2 ξ3 1
1
τ
dξ2dξ3 = 0, for all τ ∈ H, then ϕ is called a cusp form.
Remark 1. It is known that cusp forms of type (ν1, ν2) can exist only if the principal series representation parametrized by ν1, ν2 is unitary. Moreover, it is conjectured that cusp forms can exist only if ℜ(ν1) =ℜ(ν2) = 1
3.
Let Γ∞ be the group of 3×3 upper trianguler unipotent matrices with integer coefficients. Also let
Γ2 =
a b c d
1
|a, b, c, d∈Z, ad−bc=±1
, and let
Γ2∞= Γ2∩Γ∞, Γ21 ={g ∈Γ2; det(g) = 1}.
Theorem 2.1. Letϕ be a cusp form of type (ν1, ν2). There exist coefficientsam,n for positive integers n1, n2 such that
ϕ(τ) = ∑
γ∈Γ2∞\Γ2
∑∞ n1=1
∑∞ n2=1
n−11n−21an1,n2W1,1(ν1,ν2)
n1n2 n1
1
γτ
.
Proof. Let ϕ be an automorphic form of type (ν1, ν2). Since ϕ is invariant under matrices of the type
1 n3
1 n1 1
, (n1, n3 ∈Z), we may write
ϕ(τ) = ∑
n1,n2∈Z
ϕnn3
1(τ), (2.1)
where we put ϕnn3
1(τ) =
∫ 1
0
∫ 1
0
ϕ
1 ξ3 1 ξ1 1
τ
e(−n1ξ1 −n3ξ3)dξ1dξ3,
and ϕnn31 satisfies
ϕnn31
1 ξ3 1 ξ1 1
τ
=e(n1ξ1+n3ξ3)ϕnn31(τ).
Since we have
∫ 1
0
∫ 1
0
ϕ
1 ξ3 1 ξ1 1
1 n2 1
1
τ
e(−n1ξ1 −n3ξ3)dξ1dξ3
=
∫ 1
0
∫ 1
0
ϕ
1 n2 1
1
1 ξ3−n2ξ1 1 ξ1
1
τ
×e{−(n1+n2n3)ξ1 −n3(ξ3−n2ξ1)}dξ1dξ3, we have
ϕnn3
1
1 n2 1
1
τ
=ϕnn3
1+n2n3(τ), for n2 ∈Z. (2.2)
Let
( a b c d
)
∈SL(2,Z) and m∈N. We have
ϕmcmd(τ) = ϕ0m
a b c d
1
τ
. (2.3)
Because we have
∫ 1
0
∫ 1
0
ϕ
a b c d
1
1 ξ3 1 ξ1 1
τ
e(−mdξ1−mcξ3)dξ1dξ3
=
∫ 1
0
∫ 1
0
ϕ
1 bξ1+aξ3 1 dξ1 +cξ3
1
a b c d
1
τ
e{−mdξ1−mcξ3}dξ1dξ3,
By (2.1) and (2.3), we have
ϕ(τ) =ϕ00(τ) + ∑
γ∈Γ2∞\Γ21
∑∞ m=1
ϕ0m(γτ).
(2.4)
By (2.2), ϕ0m is invariant under the matrices of the form
1 n2 1
1
, (n2 ∈Z).
Thus we can write
ϕ0m(τ) =
∑∞ n=−∞
ϕm,n(τ), where ϕm,n satisfies
ϕm,n
1 ξ2 ξ3
1 ξ1 1
τ
=e(mξ1+nξ2)ϕm,n(τ).
We have
ϕm,n(τ) =
∫ 1
0
∫ 1
0
∫ 1
0
ϕ
1 ξ2 ξ3 1 ξ1 1
τ
e(−n1ξ1−n2ξ2)dξ1dξ2dξ3.
(2.4) now becomes ϕ(τ) =
∑∞ n2=−∞
ϕ0,n2(τ) + ∑
γ∈Γ2∞\Γ21
∑∞ n1=1
∑∞ n2=−∞
ϕn1,n2(γτ).
(2.5)
Let us assume now that ϕ is a cusp form. Thenϕn1,0 and ϕ0,n2 vanish. We have ϕn1,n2
−1 1
1
τ
=ϕn1,−n2(τ).
Thus
ϕ(τ) = ∑
γ∈Γ2∞\Γ2
∑∞ n1=1
∑∞ n2=1
ϕn1,n2(γτ).
Since ϕn1,n2 is a moderate growth Whittaker function, there exists an1,n2 such that ϕn1,n2(τ) =an1,n2|n1n2|−1W(ν1,ν2)
n1n2 n1
1
τ
.
Remark 2. ϕis invariant under four matrices
±1
±1 1
, hencean1,n2 =a|n1|,|n2|.
3. Jacquet’s Whittaker functions
We shall consider Whittaker functions which are defined as definite integrals, after Jacquet. Jacquet observed that these Whittaker functions had analytic continuation and functional equations, as may be deduced from the corresponding properties of the Eisenstein series, and set out to give direct proofs of these facts. We shall follow his method.
Definition 3.1. For ℜ(ν1),ℜ(ν2) > 1
3, we define the Jacquet’s Whittaker function Wk(ν1,ν2)
1,k2 (τ, w),(τ ∈ H, w ∈W) : Wk(ν1,k1,ν22)(τ, w) =
∫
n∈(N∩Pw∩Γ)\N
I(ν1,ν2)(wng)e(−k1ξ1−k2ξ2)dn,
where N and P denote the standard maximal unipotent subgroup and the standard minimal parabolic subgroup, and we put Pw =w−1P w, and ξk (k = 1,2,3) are the coordinates of n ∈N.
We have the following results by direct caliculations.
Theorem 3.1. We assume that ℜ(ν1),ℜ(ν2) > 13. The functions Wn(ν11,n,ν22)(τ, w) can be written as follows :
Wn(ν11,n,ν22)(τ, w0) =
π−3ν1−3ν2+12Γ
(3ν1
2
)
Γ
(3ν2
2
)
×Γ
(3ν1+ 3ν2−1 2
)
I(ν1,ν2)(τ), if n1 =n2 = 0,
0, otherwise,
Wn(ν11,n,ν22)(τ, w1) = W0,0(ν1,ν2)(τ, w0)e(n1x1+n2x2)
×∫∫∫
R3(ξ32+ξ22y21 +y21y22)−3ν21(ξ42+ξ12y22+y12y22)−3ν22e(−n1ξ1−n2ξ2)dξ1dξ2dξ3,
Wn(ν11,n,ν22)(τ, w2) =
W0,0(ν1,ν2)(τ, w0)e(n2x2)
×∫R(ξ22+y22)−3ν22e(−n2ξ2)dξ2, if n1 = 0,
0, otherwise.
Wn(ν11,n,ν22)(τ, w3) =
W0,0(ν1,ν2)(τ, w0)e(n1x1)
×∫R(ξ12+y12)−3ν21e(−n1ξ1)dξ1, if n2 = 0,
0, otherwise.
Wn(ν11,n,ν22)(τ, w4) =
W0,0(ν1,ν2)(τ, w0)e(n2x2)∫R(ξ32+ξ22y21 +y21y22)−3ν21
×(ξ22+y22)−3ν22e(−n2ξ2)dξ2dξ3, if n1 = 0,
0, otherwise,
Wn(ν1,ν2)
1,n2 (τ, w5) =
W0,0(ν1,ν2)(τ, w0)e(n1x1)∫R(ξ42+ξ12y22 +y21y22)−3ν22
×(ξ12+y12)−3ν21e(−n1ξ1)dξ1dξ4, if n2 = 0,
0, otherwise.
Theorem 3.2. Assume that ℜ(ν1),ℜ(ν2)> 1
3. We have the following formulae : Wn(ν11,n,ν22)(τ, w1) = |n1|ν1+2ν2−2|n2|2ν1+ν2−2W1,1(ν1,ν2)
n1n2 n1
1
τ, w1
,
Wn(ν11,0,ν2)(τ, w1) =|n1|ν1+2ν2−2W1,0(ν1,ν2)
n1 n1
1
τ, w1
,
W0,n(ν12,ν2)(τ, w1) = |n2|2ν1+ν2−2W0,1(ν1,ν2)
n2 1
1
τ, w1
,
W0,n(ν12,ν2)(τ, w2) = |n2|−ν1+ν2−1W0,1(ν1,ν2)
n2 n1
1
τ, w2
,
Wn(ν11,0,ν2)(τ, w3) = |n1|ν1−ν2−2W1,0(ν1,ν2)
n1 n1
1
τ, w3
,
W0,n(ν12,ν2)(τ, w4) = |n1|2ν1+ν2−2W0,1(ν1,ν2)
n2 1
1
τ, w4
,
Wn(ν11,0,ν2)(τ, w5) =|n1|ν1+2ν2−2W1,0(ν1,ν2)
n1 n1
1
τ, w5
.
The main results of this section is the following.
Theorem 3.3. (1) Each Wn(ν11,n,ν22)(τ, w), originally defined for ℜ(ν1),
ℜ(ν2)> 13, has meromorphic continuation to all ν1, ν2 ∈C, indeed analytic contin- uation in the case of the nondegenerate function W1,1(ν1,ν2)(τ, w).
(2) Nondegenerate Whittaker functions are entire functions of(ν1, ν2)and are in- variant under the action of the Weyl group W. By contrast, the degenerate functions are only meromorphic. They are permuted by the action of the Weyl group W.
(3) We obtain the explicit formulae in terms of the Bessel function Kν :
W1,0(ν1,ν2)(τ, w3) =2π−3ν21−3ν2+12Γ
(3ν2 2
)
Γ
(3ν1+ 3ν2−1 2
)
×y
ν1 2+ν2+12
1 yν21+2ν2e(x1)K3ν1−1 2
(2πy1),
W0,1(ν1,ν2)(τ, w2) =2π−3ν1−3ν22+12Γ
(3ν1 2
)
Γ
(3ν1+ 3ν2−1 2
)
×y1ν2+2ν1y
ν2 2+ν1+12
1 e(x2)K3ν2−1 2
(2πy2), W1,1(ν1,ν2)(τ, w1) =e(x1+x2)Wν1,ν2(y1, y2),
where Wν1,ν2(y1, y2) is defined in Theorem(1.1)–(3).
4. Eisenstein series
We will only consider the ”minimal parabolic” Eisenstein series. It is also possible to build Eisenstein series by inducing GL(2) cusp forms up toGL(3). For the latter
”maximal parabolic” Eisenstein series, we refer to Imai and Terras [I–T].
Definition 4.1. For ℜ(ν1),ℜ(ν2)> 23, τ ∈ H, the Eisenstein series E(ν1,ν2)(τ) = ∑
γ∈Γ∞\Γ
I(ν1,ν2)(γτ) is well–defined and is absolutely convergent.
We set
A1 =−a31, A2 =a22a31−a21a32, (4.1)
B1 =−a32, B2 =a21a33−a23a31, (4.2)
C1 =−a33, C2 =a23a32−a22a33, (4.3)
for
γ =
a11 a12 a13 a21 a22 a23
a31 a32 a33
∈Γ,
and set
J ={(A1, B1, C1, A2, B2, C2)| A1C2+B1B2 +C1A2 = 0,
(A1, B1, C1) = 1,(A2, B2, C2) = 1}.
Lemma 4.1. (1) By using the above notations, we have the bijection betweenΓ∞\Γ and the set J.
(2) If we associate with the matrix γ ∈ Γ the invariants (Ai, Bj, Ck) ∈ J, we obtain
I(ν1,ν2)(γτ) =I(ν1,ν2)(τ)× {(A1x3+B1x1+C1)2+ (A1x2+B1)2y12+A21y21y22}−3ν21
× {(A2x4+B2x2+C2)2+ (A2x1+B2)2y22+A22y21y22}−3ν22. Thus, by Lemma(4.1), we have
E(ν1,ν2)(τ) =I(ν1,ν2)(τ)
× ∑
(Ai,Bj,Ck)∈J
{(A1x3+B1x1+C1)2+ (A1x2+B1)2y21+A21y21y22}−3ν21
× {(A2x4+B2x2 +C2)2+ (A2x1 +B2)2y22+A22y12y22}−3ν22. The following normalization of the Eisenstein series is more convenient.
Definition 4.2. We define the Eisenstein series G(ν1,ν2)(τ) =1
4π−3ν1−3ν2+12Γ
(3ν1 2
)
Γ
(3ν2 2
)
Γ
(3ν1+ 3ν2−1 2
)
×ζ(3ν1)ζ(3ν2)ζ(3ν1+ 3ν2−1)E(ν