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Journal of Algebraic Combinatorics 8 (1998), 115–126
c 1998 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
On Elliptic Product Formulas for Jackson Integrals
Associated with Reduced Root Systems
KAZUHIKO AOMOTO
Graduate School of Mathematics, Nagoya University, Furo-cho 1, Chikusa-ku, Nagoya 464-8602, Japan
Received December 13, 1995; Revised May 5, 1997
Abstract. In this note, we state certain product formulae for Jackson integrals associated with any root systems,
involved in elliptic theta functions which appear as connection coefficients. The fomulae arise naturally in case of
arbitrary root systems by extending the connection problem which has been investigated in [1, 4] in case of A type
root system. This is also connected with the Macdonald-Morris constant term identity investigated by I. Cherednik
[6], and K. Kadell [15] on the one hand, and of the Askey-Habsieger-Kadell’s q-Selberg integral formula and its
extensions [4, 8, 12, 14, 15] on the other. This is also related with some of the results due to R.A. Gustafson [10,
11], although our integrands are different from his.
Keywords: elliptic theta function, Jackson integral, reduced root system, product formula, q-difference
1.
The product formula
Let q be the elliptic modulus satisfying |q| < 1. Let h and H be the n-dimensional Cartan
subalgebra and its Cartan subgroup, associated with a simple Lie algebra G of rank n over
C. Let h∗ be the dual of h. Let R+ ⊂ h∗ be the positive root system on h. For α, β ∈ R+ ,
we denote by (α, β) the inner product induced by the Killing form on G. We may identify
the vector space h∗ with its dual h through the inner product as follows:
α ⊂ h∗ → h α ∈ h,
such that (α, µ) = µ(h α ) for any µ ∈ h∗ .
Let X be the n-dimensional coweight lattice ∼
=Zn in h consisting of h ∈ h such that
α(h) ∈ Z. We take a suitable basis χ1 , . . . , χn , so that an arbitrary element χ of X can be
described as a linear combination
χ=
n
X
νjχj
j=1
for (ν1 , . . . , νn ) ∈ Zn . h and H are isomorphic to the tensor products X ⊗ C, and X ⊗ C∗ ,
respectively. X can also be embedded in the n-dimensional algebraic torus H (Cartan
subgroup) isomorphic to (C∗ )n . We denote this identification by
χ ∈ X → q χ = (q ν1 , . . . , q νn ) ∈ (C∗ )n .
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Let γ1 , γ2 , γ3 , . . . be arbitrary complex numbers. Each µ ∈ h∗ defines a monomial t µ
µ(χ1 )
µ(χ )
· · · tn n for t = (t1 , . . . , tn ) ∈ H. WePdenote the two linear functions λ and λ0
t1
1 P
λ = 2 α∈R+ (1 − 2γ(α,α) )α and λ0 = λ − 12 α∈R+ α, respectively.
χ
χ
=
as
We denote by Q f (t) = f (q t) the q-shift by χ ∈ X for a function f (t) on H. We
consider the q-multiplicative function 8(t) and 80 (t) on H as
Y (q 1−γ(α,α) · t α )∞
,
(q γ(α,α) · t α )∞
α∈R+
Y
(t α/2 − t −α/2 ).
80 (t) = 8(t)
8(t) = t λ
(1.1)
(1.2)
α∈R+
Q
ν
Here (x)∞ = (x; q)∞ denotes the infinite product ∞
ν=0 (1 − xq ). It is known that there
are at most 2 different γ(α,α) which appear in the RHS of (1.1).
Then 8(t), 80 (t) have the quasi-symmetry with respect to the Weyl group W associated
with the root system R+ .
σ 8(t) = 8(σ −1 (t)) = Uσ (t)8(t),
(1.3)
σ 80 (t) = 80 (σ −1 (t)) = sgn(σ )Uσ (t)8(t),
(1.4)
where {Uσ (t)}σ ∈W defines one cocycle on W with values in pseudo-constants i.e., q-periodic
functions with respect to t.
Indeed Uσ (t) satisfies the properties
Uσ τ (t) = Uσ (t) · σ Uτ (t)
for σ, τ ∈ W, Ue (t) = 1
(e = the identity)
and Q χ Uσ (t) = Uσ (t) for any χ ∈ X.
It is given in an explicit form as
½
Y
Uσ (t) =
t
(2γ(α,α) −1)α
α∈R+ ,σ (α)<0
¾
θ (q γ(α,α) t α )
,
θ (q 1−γ(α,α) t α )
(1.5)
where θ(x) denotes the Jacobi elliptic theta function θ (x) = (x)∞ · (q/x)∞ · (q)∞ .
We now consider the following Jackson integral
µ
Z
J (ξ ) =
[0,ξ ∞]q
80 (t)$,
= (1 − q)n
X
dq t1
dq tn
$ =
∧ ··· ∧
t1
tn
80 (q χ ξ ),
¶
(1.6)
χ ∈X
where ξ denotes an arbitrary point
P of H.
Assume now that λ(χ) + 12 α∈R+ α(χ) > 0 for all χ 6= 0 of the fundamental domain
1+ in h defined by the inequalities α(χ) ≥ 0 for α ∈ R+ .
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ELLIPTIC PRODUCT FORMULAS FOR JACKSON INTEGRALS
Q
Then (1.6) is summable on X ∩ 1+ because α∈R+ {(q 1−γ(α,α) t α )∞ /(q γ(α,α) t α )∞ } is bounded on 1+ . Hence, (1.6) is also summable, in view of the quasi-symmetry (1.4) of 80 (t). It
is important to note that this property does not depend on the choice of ξ , whence ξ −λ J (ξ )
becomes a meromorphic function of ξ ∈ (C∗ )n .
Then by definition J (ξ ) is a q-periodic function of ξ on H :
Q χ J (ξ ) = J (ξ )
for χ ∈ X.
The formula which we want to propose is the following.
Proposition 1
J (ξ ) can be described as
J (ξ ) = C1 ξ λ
0
Y
α∈R+
θ(qξ α )
,
θ(q 1+γ(α,α) ξ α )
(1.7)
(we shall denote by ψ̃∗ (ξ ) the term in the RHS divided by C1 in the sequel) or equivalently
X
J (ξ ) = C2
sgn σ · Uσ (ξ )−1 · σ ψ(ξ ),
(1.8)
σ ∈W
where C1 and C2 are constants with respect to ξ. Let ψ(ξ ) be a pseudo-constant defined as
ψ(ξ ) = ξ λ
0
n
Y
θ(q c j +γ(ω j ,ω j ) ξ ω j )
j=1
θ(q γ(ω j ,ω j ) ξ ω j )
,
(1.9)
P
where {c j }nj=1 are uniquely determined by the expression ω = nj=1 c j ω j with respect to
the positive simple roots {ω j }nj=1 such that the corresponding positive roots are exactly R+ .
Remark 1 Since the symmetric n-dimensional cohomology associated with the Jackson
integrals (1.6) is one dimensional, it may be conjectured that C1 and C2 can be written in
product form by using q-gamma functions of γ(α,α) . The explicit forms are presented in [13]
but not yet proved except in the two-dimensional cases. In [17], Macdonald has given a
formal proof of Ito’s formula in an entirely different way. In [10, 11] Gustafson obtains
various product formulas under a slightly different situation from ours. In [7], van Diejen
obtains a similar product formula for BCn type, by using Gustafson’s result.
Proof 1:
J (ξ ) satisfies the same quasi-symmetry as 80 (t):
σ J (ξ ) = sgn σ · Uσ (ξ ) · J (ξ ),
σ ∈W
Since J (ξ )ξ −λ is a meromorphic function on H with poles lying in the set:
ξ α ) = 0, J (ξ ) can be written as
f (ξ )
,
1+γ(α,α) ξ α )
θ(q
α∈R+
J (ξ ) = ξ λ Q
(1.10)
Q
α∈R+ θ (q
1+γ(α,α)
(1.11)
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AOMOTO
where f (ξ ) denotes a quasi-periodic function of ξ in the sense that Q χ f (ξ )/ f (ξ ) is a
monomial in ξ for any χ ∈ X, and satisfies the skew-symmetry: σ f (ξ ) = sgn σ f (ξ ) for
, . . . Hence, f (ξ ) is divided out by
σ ∈ W. ThisQimplies f (ξ ) vanishes if ξ α = 1, q ±1 , q ±2
0 Q
the product α∈R+ θ(qξ α ) and also by the product ξ λ α∈R+ θ (qξ α ):
0
J (ξ ) = ξ λ g(ξ )
Y
α∈R+
θ(qξ α )
,
θ(q 1+γ(α,α) ξ α )
(1.12)
where g(ξ ) denotes a holomorphic function on H. Since g(ξ ) is q-periodic, it must be
constant. And the formula (1.7) follows.
P
To prove (1.8), we denote by ψ̃(ξ ) the sum σ ∈W sgn σ Uσ−1 (ξ ) · σ ψ(ξ ). Since Uσ (t)
Q
has the expression (1.5), the poles of ψ̃(ξ ) lie in the set {ξ ∈ H ; α∈R+ θ (q 1+γ(α,α) ξ α ) = 0}.
Moreover, it satisfies the same quasi-symmetry as in (1.10). Hence, it can be expressed as
in the RHS of (1.8). It is sufficient to prove that it does not vanish identically. We evaluate
the residues of ψ̃(ξ ) at the points of the equations θ (q γ(ω j ,ω j ) ξ ω j ) = 0, say, of the equations
q γ(ω j ,ω j ) ξ ω j = 1
for 1 ≤ j ≤ n.
(1.13)
These points appear only for the residues of the summand ψ(ξ ) itself in the sum ψ̃(ξ ).
In fact, σ (ω j ) = ω j , 1 ≤ j ≤ n if and only if σ = the identity. Since they do not vanish
identically, ψ̃(ξ ) does not vanish identically. Let ξ = ζ be a point satisfying this system of
equations. Then
Qn
Resξ =ζ ψ̃(ξ ) = Resξ =ζ ψ(ξ ) = ζ
j=1 θ (q
θ 0 (1)n
λ0
cj
)
.
2
Equation (1.8) is thus proved.
2.
A-type root system
In the sequel ν1 , ν2 , ν3 , . . . will denote integers.
At first we can take 8(t) in a slightly more general way than (1.1), namely we take 8(t)
as
8(t) = t1α1 · · · tnαn
n
Y
(t j )∞
(q β t j )∞
j=1
Y
1≤i< j≤n
(q 1−γ t j /ti )∞
(q γ t j /ti )∞
(2.1)
for α j , β, γ ∈ C such that α j = α1 + ( j − 1)(1 − 2γ ), and
80 (t) = 8(t)D(t),
for D(t) =
Y
1≤i< j≤n
(ti − t j ).
(2.2)
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119
X consists of the lattice points x = (ν1 , . . . , νn ) ∈ Zn . In this case, by the same argument
as in the preceding section, the sum (1.6) is summable provided
−β > α1 + n − 1 > 0,
−β > α1 + (n − 1)(2 − 2γ ) > 0.
The Weyl group W is isomorphic to the symmetric group Sn of nth degree and the
one-cocycle {Uσ (t)} is given as
µ ¶2γ −1
Y
θ (q γ t j /ti )
tj
Uσ (t) =
(2.3)
ti
θ (q 1−γ t j /ti )
i< j,σ −1 (i)>σ −1 ( j)
for any permutation σ among the n letters {1, 2, . . . , n}.
As a special case of Jackson integrals, we take ξ = ξ F for
ξ1 = q,
ξ2 = q 1+γ , . . . , ξn = q 1+(n−1)γ .
The integral J over [0, ξ F ∞]q is done only over the set hξ F i consisting of the points t such
that t1 = q 1+ν1 , t2 /t1 = q γ +ν2 , . . . , tn /tn−1 = q γ +νn for each ν j ∈ Z≥0 . hξ F i is clearly a
subset of [0, ξ F ∞]q . We call it “α-stable cycle”, since the absolute value |t α | is maximal
at ξ F and smaller elsewhere in hξ F i than the value at ξ F . (Here the terminology “stable” is
used as a discrete analog of the one in case of ordinary integrals.)
Hence, the substitution of the point ξ F into 80 (t) gives the asymptotic behavior of
J = J (α1 ) for α1 → +∞ as
J = q An
n
Y
0q (β + 1 + ( j − 1)γ )0q ( jγ )
(1 + O(q α1 ))
0q (γ )
j=1
(2.4)
P
for An = nj=1 (α j + n − j)[1 + ( j − 1)γ ], where 0q (u) denotes the q-gamma function
(1 − q)1−u (q)∞ /(q u )∞ .
On the other hand, we can take ξ = η F for
ξ1 = q −β ,
ξ2 = q −β−γ , . . . , ξn = q −β−(n−1)γ .
The Jackson integral J over [0, η F ∞]q is meaningless, since 80 (t) has poles there. We
denote by hη F i the set of points t such that
t1 = q −β−ν1 ,
t2 /t1 = q −γ −ν2 , . . . , tn /tn−1 = q −γ −νn
for ν j ∈ Z≥0 . hη F i is a subset of [0, η F ∞]q . Then we can replace the sum (1.6) by the
following regularization (see [1] or [4] for details):
µ
¶
Z
Z
80 (t) · $
also denoted by
80 (t) · $
reg
=
[0,η F ∞]q
n
XY
ν j ≥0 j=1
·
Rest j /t j−1 =q −γ −ν j
reghηF i
¸
dt1
dtn
∧ ··· ∧
80 (t)
,
t1
tn
(2.5)
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Q
where nj=1 Res denotes the residue at each point of hη F i. Here we have put t0 = q −β+γ .
Equation (2.5) is well defined because 80 (t) has simple poles at each point of hη F i.
We define the quotient of theta functions
ψn (ξ, η F ) = (1 − q)n
n
Y
¡
ξ j q β+( j−1)γ
j=1
¢α j (q)3∞ θ (q α j +···+αn +γ +1 ξ j /ξ j−1 )
θ (q α j +···+αn +1 )θ (q γ +1 ξ j /ξ j−1 )
(2.6)
for ξ0 = q −β+γ . It is a pseudo constant. The connection coefficient (or ratio) between the
Jackson integrals over the cycles [0, ξ ∞]q and reghη F i is denoted by ([0, ξ ∞]q : reghη F i)80 .
Then we can give the following formula (see [4]):
([0, ξ ∞]q : reghη F i)80 =
X
σ ψn (ξ, η F ) · sgn σ · Uσ (ξ )−1 ,
(2.7)
σ ∈Sn
where σ ϕ(ξ ) denotes ϕ(σ −1 (ξ )).
We evaluate the RHS of (2.7) in case where β vanishes. We denote by Fn (ξ ) = Fn (ξ, α1 , γ )
the sum
X
(
−1
sgn σ · Uσ (ξ )
·σ
σ ∈Sn
n
Y
α
ξj j
j=1
θ (q α j +···+αn +γ +1 ξ j /ξ j−1 )
θ (q γ +1 ξ j /ξ j−1 )
)
(2.8)
for ξ0 = q γ . Then we have the product formula (see [2]):
Fn (ξ, α1 , γ ) = f n (α1 , γ ) · gn (ξ, α1 , γ ),
(2.9)
where
f n (α1 , γ ) = (−1)n(n−1)/2
n
Y
θ (q α j +···+αn +1 )
,
θ (q 1+α1 −(n+ j−2)γ )
(2.10)
Y
θ (q α1 +1−(n−1)γ ξ j )
ξi θ (ξ j /ξi )
·
.
θ (qξ j )
θ (q γ +1 ξ j /ξi )
1≤i< j≤n
(2.11)
q −( j−1)
2
j=1
gn (ξ, α1 , γ ) =
n
Y
j=1
α −2( j−1)γ
ξj 1
γ
Hence, ξ j ( j ≥ 1) being replaced by ξ j q β in (2.9),
([0, ξ ∞]q : reghη F i)80
n
Y
(q)3∞ q α j γ ( j−1)
· f n (α1 , γ ) · gn (q β ξ, α1 , γ ).
= (1 − q)n
α j +···+αn +1 )
θ(q
j=1
It is also possible to evaluate explicitly
is known (see [16]).
R
hξ F i 80 (t)·$
(2.12)
itself. In fact, the following formula
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ELLIPTIC PRODUCT FORMULAS FOR JACKSON INTEGRALS
Proposition 2
Z
hξ F i
80 (t) · $
= q An
n
Y
0q (β + 1 + ( j − 1)γ )0q (α1 + n − 1 − (n + j − 2)γ )0q ( jγ )
.
0q (γ )0q (α1 + β + n − (n − j)γ )
j=1
In particular, we have
(hξ F i : reg hη F i)80
Kn
= (1 − q)n (q)3n
·
∞ ·q
n
Y
θ (q α1 +β+2−(n− j)γ )θ (q γ )
,
θ (q α1 +1−(n+ j−2)γ )θ (q β+2+( j−1)γ )θ (q jγ )
j=1
(2.13)
where K n denotes
2
n(n − 1)
γ + nβ{α1 − (n − 1)γ }
K n = − n(n − 1)(2n − 1)γ 2 −
3
2
+ nα1 {1 + (n − 1)γ }.
(2.14)
Next, we take as 8(t) the q-multiplicative function
8(t) = t1α1 · · · tnαn
(t1 · · · tn )∞
(q β t1 · · · tn )∞
Y
1≤i< j≤n
(q 1−γ t j /ti )∞
(q γ t j /ti )∞
(2.15)
for α j = α1 + ( j − 1)(1 − 2γ ). Let 80 (t) be as in (2.2). As in case of (2.1), J (ξ ) is
summable, provided
−β > α1 + n − 1 > 0,
−β > α1 + (n − 1)(2 − 2γ ) > 0.
X consists of the points x = (ν1 , . . . , νn ) ∈ Zn . Then by the argument in Section 1, we
have the formula (1.7) for
J (ξ ) = C1
n
Y
α −2( j−1)γ
ξj 1
j=1
3.
Y
θ (qξ j /ξi )
θ (qξ1 · · · ξn )
.
1+β
θ(q ξ1 · · · ξn ) 1≤i< j≤n θ (q 1+γ ξ j /ξi )
(2.16)
Bn , Cn , Dn , G2 , F4 and E6 ∼ E8 type root systems
For Bn -type, we take as 8(t) the multiplicative function
8(t) = t1α1 · · · tnαn
n
Y
(t j q 1−β )∞
j=1
(t j
qβ )
∞
Y
1≤i< j≤n
(q 1−γ t j /ti )∞ · (q 1−γ ti t j )∞
(q γ t j /ti )∞ · (q γ ti t j )∞
(3.1)
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for α j = α1 + ( j − 1)(1 − 2γ ), α1 = 12 − β. This is exactly the one obtained from (1.1)
for the root system Bn when an explicit realization of positive roots is properly chosen for
β = γ1 , γ = γ2 . The Weyl group W is isomorphic to the semi-direct product Zn2 × Sn .
80 (t) is given by
¡ −(1/2)+n −(3/2)+n
¢
t2
· · · tn(1/2)
80 (t) = 8(t)A t1
n
Y
= (−1)n(n+1)/2 8(t)(t1 · · · tn )−n+(1/2) (1 − t j )
Y
·
j=1
(ti − t j )(1 − ti t j ),
(3.2)
1≤i< j≤n
where A denotes the alternating sum with respect to the Weyl group W.
. . . , νn ) ∈ Zn .α
X consists of the points x = (ν1 , Q
θ (qξ )
λ0
Let ψ̃∗ (ξ ) denote the product ξ
α∈R+ θ (q 1+γ(α,α) ξ α ) in the RHS of (1.7). Then ψ̃∗ (ξ ) is
represented more concretely as
ψ̃∗ (ξ ) =
n
Y
α j −( j− 12 )
ξj
j=1
Y
θ(qξ j )
θ (qξ j /ξi )θ (qξi ξ j )
.
β+1
θ(q ξ j ) 1≤i< j≤n θ (q 1+γ ξ j /ξi )θ (q 1+γ ξi ξ j )
The explicit formula for C1 has given by Ito [13] in case n = 2.
Similarly for Cn -type, we take as 8(t) the function
¡ 1−β 2 ¢
n
Y
q t j ∞ Y (q 1−γ t j /ti )∞ · (q 1−γ ti t j )∞
α1
αn
¡ 2¢
8(t) = t1 · · · tn
q β t j ∞ 1≤i< j≤n (q γ t j /ti )∞ · (q γ ti t j )∞
j=1
(3.3)
(3.4)
for α1 = 1 − 2β, α j = α1 + ( j − 1)(1 − 2γ ). This corresponds to (1.1) for γ = γ2 , β = γ4 .
The Weyl group W is isomorphic to the one of Bn -type. 80 (t) is equal to 8(t)A(t1n t2n−1 · · · tn ).
X consists of the points x = (ν1 , . . . , νn ) ∈ Zn or (ν1 + 12 , . . . , νn + 12 ) ∈ Zn +( 12 , . . . , 12 ).
ψ̃∗ (ξ ) is then given as
¡ 2¢
n
Y
Y
θ (qξ j /ξi )θ (qξi ξ j )
α j − j θ qξ j
¡
¢
ξj
.
(3.5)
ψ̃∗ (ξ ) =
1+γ ξ /ξ )θ (q 1+γ ξ ξ )
β+1 ξ 2
θ
(q
θ
q
j i
i j
j 1≤i< j≤n
j=1
For Dn -type we take as 8(t) the function
8(t) =
n
Y
α − j+1
tj j
j=1
Y
1≤i< j≤n
(q 1−γ t j /ti )∞ · (q 1−γ ti t j )∞
(q γ t j /ti )∞ · (q γ ti t j )∞
(3.6)
for α1 = 0 and α j = α1 + ( j − 1)(1 − 2γ ). This corresponds to (1.1) for γ = γ2 . 80 (t) is
equal to 8(t)A(t1n t2n−1 · · · tn ). The Weyl group W is isomorphic to the semi-direct product
of Zn2 and Sn . X is the same as in case of Cn type. ψ̃∗ (ξ ) is then equal to
ψ̃∗ (ξ ) =
n
Y
j=1
α −( j−1)
ξj j
Y
1≤i< j≤n
θ (qξ j /ξi )θ (qξi ξ j )
.
θ (q 1+γ ξ j /ξi )θ (q 1+γ ξi ξ j )
(3.7)
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For E 8 -type we can choose an orthonormal basis {² j }8j=1 in h∗ ∼
= R8 such that the positive
simple roots are
1
(²1 + ²8 − ²2 − ²3 − ²4 − ²5 − ²6 − ²7 ),
2
ω2 = ²1 + ²2 , ω3 = ²2 − ²1 , ω4 = ²3 − ²2 , ω5 = ²4 − ²3 , ω6 = ²5 − ²4 ,
ω7 = ²6 − ²5 , ω8 = ²7 − ²6 .
P7
The positive roots are ±²i + ² j (1 ≤ i < j ≤ 8) and 12 (²8 + i=1
ν(i)²i ) where {ν(k)}7k=1
moves over the set ±1 such that the number of ν(k) which are equal to −1 is even. Then
8(t) has the quasi-symmetry when and only when
ω1 =
α j = ( j − 1)(1 − 2γ )(1 ≤ j ≤ 7), α8 = 23(1 − 2γ ).
8
Y
Y (q 1−γ t j /ti )∞ (q 1−γ ti t j )∞
α
8(t) =
tj j
(q γ t j /ti )∞ (q γ ti t j )∞
j=1
1≤i< j≤8
¡ 1−γ ν(1)/2
ν(7)/2 1/2 ¢
· · · t7
t8 ∞
q t1
·
¡ ν(1)/2
¢ .
ν(7)/2
1/2
· · · t7
t8 ∞
q γ t1
ν(k)=±1
Y
The pairing between µ =
µ, x →
8
X
P8
j=1
(3.8)
µ j ² j ∈ h∗ and x = (x1 , . . . , x8 ) ∈ h(∼
= R8 ) is given by
µjxj.
j=1
X is the eight-dimensional lattice in h consisiting of the points x = (x1 , . . . , x6 , x7 , x8 )
∈ h such that α(x) ≡ 0(Z) for α ∈ R+ . In other words, x ∈ X consists of the points x such
that x j = ν j , (1 ≤ j ≤ 8) or x j = ν j + 12 , (1 ≤ j ≤ 8) satisfying the equality
ν1 + · · · + ν8 ≡ 0(2).
ψ̃∗ (ξ ) is then equal to
(
ψ̃∗ (ξ ) =
7
Y
)
−2( j−1)γ
ξj
j=1
·
−46γ
ξ8
Y
θ (qξ j /ξi )θ (qξi ξ j )
θ (q 1+γ ξ j /ξi )θ (q 1+γ ξi ξ j )
1≤i< j≤8
¡ ν(1)/2
ν(7)/2 1/2 ¢
θ qξ1
· · · ξ7
ξ8
.
¡
ν(1)/2
ν(7)/2 1/2 ¢
γ
+1
ξ1
· · · ξ7
ξ8
ν(k)=±1 θ q
Y
(3.9)
For E 7 -type, in terms of the above basis {² j }8j=1 , the positive simple roots are
1
(²1 + ²8 − ²2 − ²3 − ²4 − ²5 − ²6 − ²7 ),
2
ω2 = ²1 + ²2 , ω3 = ²2 − ²1 , ω4 = ²3 − ²2 , ω5 = ²4 − ²3 ,
ω7 = ²6 − ²5 .
ω1 =
ω6 = ²5 − ²4 ,
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P6
The positive roots are ±²i + ² j (1 ≤ i < j ≤ 6), −²7 + ²8 and 12 (−²7 + ²8 + i=1
ν(i)²i )
6
where {ν(k)}k=1 moves over the set ±1 such that the number of ν(k) which are equal to −1
is odd. Then 8(t) has the quasi-symmetry when and only when
17
α j = ( j − 1)(1 − 2γ )(1 ≤ j ≤ 6), −α7 = α8 =
(1 − 2γ ).
2 )
(
(
)
6
Y (q 1−γ t j /ti )∞ (q 1−γ ti t j )∞
Y
(q 1−γ t8 )∞
α
t j j t8α8 ·
8(t) =
·
γ
γ
(q t j /ti )∞ (q ti t j )∞
(q γ t8 )∞
j=1
1≤i< j≤6
¡ 1−γ ν(1)/2
ν(6)/2 1/2 ¢
· · · t6
t8 ∞
q t1
·
.
¡ ν(1)/2
ν(6)/2 1/2 ¢
γ
· · · t6
t8 ∞
q t1
ν(k)=±1
Y
(3.10)
P
X consists of the points x such that either x j = ν j (1 ≤ j ≤ 8), x7 = 0 where 6j=1 ν j +
P
6
ν8 ≡ 0(2), or x j = ν j + 12 (1 ≤ j ≤ 6), x7 = 0, x8 = ν8 where j=1 ν j + ν8 ≡ 1(2).
We normalize ξ as ξ7 = 1. ψ̃∗ (ξ ) is then equal to
(
)
6
Y
Y
θ (qξ j /ξi )θ (qξi ξ j )
θ (qξ8 )
−2( j−1)γ
−17γ
ξj
·
· ξ8
ψ̃∗ (ξ ) =
1+γ ξ /ξ )θ (q 1+γ ξ ξ ) θ (q γ +1 ξ )
θ
(q
j i
i j
8
j=1
1≤i< j≤6
¡ ν(1)/2
ν(6)/2 1/2 ¢
θ qξ1
· · · ξ6
ξ8
·
.
¡
ν(1)/2
ν(6)/2
1/2 ¢
γ +1 ξ
· · · ξ6
ξ8
ν(k)=±1 θ q
1
Y
(3.11)
For E 6 -type, in R8 the positive simple roots are
1
(²1 + ²8 − ²2 − ²3 − ²4 − ²5 − ²6 − ²7 ),
2
ω2 = ²1 + ²2 , ω3 = ²2 − ²1 , ω4 = ²3 − ²2 , ω5 = ²4 − ²3 ,
ω1 =
ω6 = ²5 − ²4 .
P5
ν(i)²i )
The positive roots are ±²i + ² j (1 ≤ i < j ≤ 5) and 12 (²8 − ²6 − ²7 + i=1
where {ν(k)}6k=1 moves over the set ±1 such that the number of ν(k) which are equal to −1
is even. Then 8(t) has the quasi-symmetry when and only when
α j = ( j − 1)(1 − 2γ )(1 ≤ j ≤ 5), −α6 = −α7 = α8 = 4(1 − 2γ ).
(
(
)
)
5
Y (q 1−γ t j /ti )∞ (q 1−γ ti t j )∞
Y
αj
α8
8(t) =
t j t8 ·
(q γ t j /ti )∞ (q γ ti t j )∞
j=1
1≤i< j≤5
¡ 1−γ ν(1)/2
ν(5)/2 1/2 ¢
· · · t5
t8 ∞
q t1
·
¡ ν(1)/2
¢ .
ν(5)/2
1/2
· · · t5
t8 ∞
q γ t1
ν(k)=±1
Y
(3.12)
X consists of the points x such that either x j = ν j (1 ≤ j ≤ 8), x6 = x7 = 0 where
P5
ν j + ν8 ≡ 0(2), or x j = ν j + 12 (1 ≤ j ≤ 5), x6 = x7 = 0, x8 = ν8 − 12 where
P5j=1
j=1 ν j + ν8 ≡ 1(2).
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ELLIPTIC PRODUCT FORMULAS FOR JACKSON INTEGRALS
125
We normalize ξ as ξ6 = ξ7 = 1. ψ̃∗ (ξ ) is then equal to
(
5
Y
ψ̃∗ (ξ ) =
)
−2( j−1)γ
ξj
−8γ
· ξ8
j=1
Y
θ (qξ j /ξi )θ (qξi ξ j )
1+γ ξ /ξ )θ (q 1+γ ξ ξ )
θ
(q
j i
i j
1≤i< j≤5
¡ ν(1)/2
ν(5)/2 1/2 ¢
θ qξ1
· · · ξ5
ξ8
·
.
¡
ν(1)/2
ν(5)/2 1/2 ¢
γ
+1
ξ1
· · · ξ5
ξ8
ν(k)=±1 θ q
Y
(3.13)
For F4 -type we can choose an orthonormal basis {(² j )}4j=1 such that the positive simple
roots are ω1 = ²2 − ²3 , ω2 = ²3 − ²4 , ω3 = ²4 , ω4 = 12 (²1 − ²2 − ²3 − ²4 ). The positive
roots are ² j , ²i ± ² j (1 ≤ i < j ≤ 4) and 12 (²1 ± ²2 ± ²3 ± ²4 ). We put
11
8(t) = t12
−5β−6β
5
t22
−β−4γ
3
t32
−β−2γ
1
t42
−β
·
4
Y
(q 1−β t j )∞
j=1
(q β t j )∞
¡
¢
Y q 1−β t11/2 t2ν(2)/2 t3ν(3)/2 t4ν(4)/2
(q 1−γ ti /t j )∞ (q 1−γ ti t j )∞
∞
·
·
¡ 1/2 ν(2)/2 ν(3)/2 ν(4)/2 ¢ ,
γ t /t ) (q γ t t )
βt
(q
i
j
∞
i
j
∞
t
t
t
q
1≤i< j≤4
ν(k)=±1
1
2
3
4
∞
Y
(3.14)
where ν(k) moves over the set {±1}.
X consists of the points x = (x1 , x2 , x3 , x4 ) such that x j = ν j where ν1 + ν2 + ν3 + ν4 ≡
0(2).
ψ̃∗ (ξ ) is then equal to
−5β−6γ −β−4γ −β−2γ −β
ξ2
ξ3
ξ4
ψ̃∗ (ξ ) = ξ1
4
Y
θ (qξi )
θ (q 1+β ξi )
j=1
¡ 1/2 ν(2)/2 ν(3)/2 ν(4)/2 ¢
Y
θ qξ1 ξ2
ξ3
ξ4
θ(qξi /ξ j )θ (qξi ξ j )
·
·
.
¡
1/2
ν(2)/2
ν(3)/2
ν(4)/2 ¢
1+γ
1+γ
θ(q ξi /ξ j )θ (q ξi ξ j ) ν(k)=±1 θ q 1+β ξ1 ξ2
ξ3
ξ4
1≤i< j≤4
Y
(3.15)
For G 2 -type we can choose an orthonormal basis ²1 , ²2 , ²3 such that the positive simple
roots are ²1 − ²2 , −2²1 + ²2 + ²3 and that the positive roots are ²1 − ²2 , −²2 + ²3 , −²1 +
²3 , ²1 + e3 − 2²2 , ²2 + ²3 − 2²1 , 2²3 − ²1 − ²2 . 8(t) is given by
(q 1−β t1 )∞ (q 1−β t3 )∞ (q 1−β t3 /t1 )∞
(q β t1 )∞ (q β t3 )∞ (q β t3 /t1 )∞
¡ 1−γ ± 2 ¢ ¡ 1−γ 2 ± ¢
q t3 t1 ∞ q t3 t1 ∞ (q 1−γ t1 t3 )∞
¡
± ¢ ¡
± ¢
×
.
q γ t3 t12 ∞ q γ t32 t1 ∞ (q γ t1 t3 )∞
2γ −1 3−2β−4γ
t3
8(t) = t1
(3.16)
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AOMOTO
X consists of the points x = (x1 , x2 , x3 ) such that x1 = ν1 , x3 = ν3 , x2 = 0. We normalize
ξ as ξ2 = 1. ψ̃∗ (ξ ) is then equal to
±
θ(qξ1 )θ (qξ3 )θ (qξ3 ξ1 )
ψ̃∗ (ξ ) =
θ(q β+1 ξ1 )θ (q β+1 ξ3 )θ (q β+1 ξ3 ξ1 )
¡ ± 2¢ ¡ 2± ¢
θ qξ3 ξ θ qξ ξ1 θ (qξ1 ξ3 )
.
× ¡ γ +1 ± 2 ¢1 ¡ γ +13 2 ± ¢ γ +1
θ q ξ3 ξ1 θ q ξ3 ξ1 θ (q ξ1 ξ3 )
2γ −2β−4γ
ξ1 ξ3
(3.17)
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