On Elliptic Product Formulas for Jackson Integrals Associated with Reduced Root Systems

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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 algebraGof rank n over C. Let h^∗be the dual of h.Let R₊ ⊂h^∗be the positive root system on h.Forα, β ∈ R₊, we denote by(α, β)the inner product induced by the Killing form onG. 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 anyµ∈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 basisχ1, . . . , χn,so that an arbitrary elementχof X can be described as a linear combination

χ= Xn

j=1

νjχj

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.

Letγ1, γ2, γ3, . . .be arbitrary complex numbers. Eachµ∈h^∗defines a monomial t^µ= t₁^µ(χ1)· · ·tn^µ(χn)for t =(t1, . . . ,tn) ∈ H.We denote the two linear functionsλandλ⁰ as λ= ¹₂P

α∈R₊(1−2γ_(α,α))αandλ⁰=λ−¹₂P

α∈R₊α, respectively.

We denote by Q^χf(t)= f(q^χt)the q-shift byχ ∈ X for a function f(t)on H.We consider the q-multiplicative function8(t)and80(t)on H as

8(t)=t^λ Y

α∈R₊

(q^{1−γ(α,α)}·t^α)_∞

(q^γ(α,α)·t^α)_∞ , (1.1)

80(t)=8(t) Y

α∈R₊

(t^α/2−t^−α/2). (1.2)

Here(x)∞=(x;q)∞denotes the infinite productQ_∞

ν=0(1−xq^ν).It is known that there are at most 2 differentγ(α,α)which appear in the RHS of (1.1).

Then8(t), 80(t)have the quasi-symmetry with respect to the Weyl group W associated with the root system R₊.

σ8(t)=8(σ⁻¹(t))=U_σ(t)8(t), (1.3)

σ 80(t)=80(σ⁻¹(t))=sgn(σ )U_σ(t)8(t), (1.4) where{U_σ(t)}σ∈Wdefines one cocycle on W with values in pseudo-constants i.e., q-periodic functions with respect to t .

Indeed U_σ(t)satisfies the properties

U_{σ τ}(t)=U_σ(t)·σU_τ(t) forσ, τ ∈W,Ue(t)=1 (e=the identity) and Q^χU_σ(t)=U_σ(t)for anyχ∈ X.

It is given in an explicit form as U_σ(t)= Y

α∈R₊,σ (α)<0

½

t^{(2γ(α,α)−1)α} θ(q^γ(α,α)t^α) θ(q^{1−γ(α,α)}t^α)

¾

, (1.5)

whereθ(x)denotes the Jacobi elliptic theta functionθ(x)=(x)_∞·(q/x)_∞·(q)_∞. We now consider the following Jackson integral

J(ξ)= Z

[0,ξ∞]_q

80(t)$, µ

$ =d_qt₁

t₁ ∧ · · · ∧d_qt_n t_n

¶

=(1−q)ⁿX

χ∈X

80(q^χξ), (1.6)

whereξ denotes an arbitrary point of H. Assume now thatλ(χ)+¹₂P

α∈R₊α(χ) >0 for allχ 6=0 of the fundamental domain 1₊in h defined by the inequalitiesα(χ)≥0 forα∈ R₊.

Then (1.6) is summable on X∩1+becauseQ

α∈R₊{(q^{1−γ(α,α)}t^α)∞/(q^γ(α,α)t^α)∞}is bound- ed on1+.Hence, (1.6) is also summable, in view of the quasi-symmetry (1.4) of80(t). It is important to note that this property does not depend on the choice ofξ, whenceξ^−λJ(ξ) becomes a meromorphic function ofξ ∈(C^∗)ⁿ.

Then by definition J(ξ)is a q-periodic function ofξ on H : Q^χJ(ξ)=J(ξ) forχ∈ X.

The formula which we want to propose is the following.

Proposition 1 J(ξ)can be described as J(ξ)=C1ξ^λ0 Y

α∈R₊

θ(qξ^α)

θ(q^1+γ(α,α)ξ^α), (1.7)

(we shall denote byψ˜∗(ξ)the term in the RHS divided by C1in the sequel) or equivalently J(ξ)=C2

X

σ∈W

sgnσ·U_σ(ξ)⁻¹·σ ψ(ξ), (1.8)

where C₁and C₂are constants with respect toξ.Letψ(ξ)be a pseudo-constant defined as ψ(ξ)=ξ^λ0

Yn j=1

θ(q^{cj+γ(ωj,ωj)}ξ^ωj)

θ(q^γ(ωj,ωj)ξ^ωj) , (1.9)

where{cj}ⁿj=1 are uniquely determined by the expressionω=Pn

j=1cjωj with respect to the positive simple roots{ωj}ⁿj=1such 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 C2can be written in product form by using q-gamma functions ofγ_(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 BC_ntype, by using Gustafson’s result.

Proof 1: J(ξ)satisfies the same quasi-symmetry as80(t):

σJ(ξ)=sgnσ·U_σ(ξ)·J(ξ), σ ∈W (1.10)

Since J(ξ)ξ^−λis a meromorphic function on H with poles lying in the set:Q

α∈R₊θ(q^1+γ(α,α) ξ^α)=0,J(ξ)can be written as

J(ξ)=ξ^λQ f(ξ)

α∈R₊θ(q^1+γ(α,α)ξ^α), (1.11)

where f(ξ) denotes a quasi-periodic function ofξ in the sense that Q^χf(ξ)/f(ξ) is a monomial inξ for anyχ ∈ X,and satisfies the skew-symmetry: σf(ξ) =sgnσf(ξ)for σ ∈W.This implies f(ξ)vanishes ifξ^α =1,q^±1,q^±2, . . .Hence, f(ξ)is divided out by the productQ

α∈R₊θ(qξ^α)and also by the productξ^λ0Q

α∈R₊θ(qξ^α): J(ξ)=ξ^λ0g(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.

To prove (1.8), we denote byψ(ξ)˜ the sumP

σ∈WsgnσU_σ⁻¹(ξ)·σψ(ξ).Since U_σ(t) has the expression (1.5), the poles ofψ(ξ)˜ lie in the set{ξ ∈ H;Q

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 ofψ(ξ)˜ at the points of the equationsθ(q^γ(ωj,ωj)ξ^ωj)=0, say, of the equations

q^γ(ωj,ωj)ξ^ωj =1 for 1≤ j ≤n. (1.13)

These points appear only for the residues of the summandψ(ξ)itself in the sumψ(ξ).˜ In fact,σ(ωj)=ωj,1 ≤ j ≤ n if and only ifσ=the identity.Since they do not vanish identically,ψ(ξ)˜ does not vanish identically. Letξ =ζ be a point satisfying this system of equations. Then

Res_ξ=ζψ(ξ)˜ =Res_ξ=ζψ(ξ)=ζ^λ0 Qn

j=1θ(q^cj) θ⁰(1)ⁿ .

Equation (1.8) is thus proved. 2

2. A-type root system

In the sequelν1, ν2, ν3, . . .will denote integers.

At first we can take8(t)in a slightly more general way than (1.1), namely we take8(t) as

8(t)=t₁^α1· · ·t_n^αn Yn

j=1

(t_j)_∞ (q^βt_j)_∞

Y

1≤i<j≤n

(q^1−γt_j/t_i)_∞

(q^γt_j/t_i)_∞ (2.1)

forαj, β, γ ∈C such thatαj=α1+(j−1)(1−2γ ), and 80(t)=8(t)D(t), for D(t)= Y

1≤i<j≤n

(ti−tj). (2.2)

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{U_σ(t)}is given as

U_σ(t)= Y

i<j,σ⁻¹(i)>σ⁻¹(j)

µt_j t_i

¶2γ−1 θ(q^γt_j/t_i)

θ(q^1−γt_j/t_i) (2.3)

for any permutationσ among the n letters{1,2, . . . ,n}.

As a special case of Jackson integrals, we takeξ =ξF for ξ1=q, ξ2 =q^1+γ, . . . , ξn =q^1+(n−1)γ.

The integral J over [0, ξF∞]_q is done only over the sethξFiconsisting of the points t such that t1 =q^1+ν1,t2/t1 =q^γ+ν2, . . . ,tn/tn−1 =q^γ+νn for eachνj ∈ Z_≥0.hξFiis clearly a subset of [0, ξF∞]q.We call it “α-stable cycle”, since the absolute value|t^α|is maximal atξFand smaller elsewhere inhξFithan the value atξF.(Here the terminology “stable” is used as a discrete analog of the one in case of ordinary integrals.)

Hence, the substitution of the point ξF into 80(t)gives the asymptotic behavior of J =J(α1)forα1→ +∞as

J =q^An Yn

j=1

0q(β+1+(j−1)γ )0q(jγ )

0q(γ ) (1+O(q^α1)) (2.4)

for An =Pn

j=1(αj+n− j)[1+(j−1)γ], where0q(u)denotes the q-gamma function (1−q)^1−u(q)_∞/(q^u)_∞.

On the other hand, we can takeξ =ηF for

ξ1=q^−β, ξ2=q^−β−γ, . . . , ξn =q^{−β−(n−1)γ}.

The Jackson integral J over [0, ηF∞]q is meaningless, since80(t)has poles there. We denote byhηFithe set of points t such that

t1=q^−β−ν1, t2/t1=q^−γ−ν2, . . . ,tn/tn−1=q^−γ−νn

forνj ∈ Z_≥0.hηFi is a subset of [0, ηF∞]q.Then we can replace the sum (1.6) by the following regularization (see [1] or [4] for details):

reg Z

[0,ηF∞]_q

80(t)·$ µ

also denoted by Z

reghηFi80(t)·$

¶

=X

νj≥0

Yn j=1

Res_tj/tj−1=q−γ−νj

·

80(t)dt1

t1

∧ · · · ∧dtn

tn

¸

, (2.5)

whereQn

j=1Res denotes the residue at each point ofhηFi. Here we have put t₀=q^−β+γ. Equation (2.5) is well defined because80(t)has simple poles at each point ofhηFi.

We define the quotient of theta functions ψn(ξ, ηF)=(1−q)ⁿ

Yn j=1

¡ξjq^β+(j−1)γ¢_αj (q)³_∞θ(q^{αj+···+αn+γ+1}ξj/ξj−1)

θ(q^{αj+···+αn+1})θ(q^γ+1ξj/ξj−1) (2.6) forξ0=q^−β+γ. It is a pseudo constant. The connection coefficient (or ratio) between the Jackson integrals over the cycles [0, ξ∞]qand reghηFiis denoted by([0, ξ∞]q: reghηFi)80.

Then we can give the following formula (see [4]):

([0, ξ∞]q : reghηFi)80=X

σ∈Sn

σψn(ξ, ηF)·sgnσ·U_σ(ξ)⁻¹, (2.7)

whereσϕ(ξ)denotesϕ(σ⁻¹(ξ)).

We evaluate the RHS of (2.7) in case whereβvanishes. We denote by Fn(ξ)=Fn(ξ, α1, γ ) the sum

X

σ∈Sn

sgnσ·U_σ(ξ)⁻¹·σ ( _n

Y

j=1

ξ^α_j^jθ(q^{αj+···+αn+γ+1}ξj/ξj−1) θ(q^γ+1ξj/ξj−1)

)

(2.8)

forξ0 =q^γ.Then we have the product formula (see [2]):

Fn(ξ, α1, γ )= fn(α1, γ )·gn(ξ, α1, γ ), (2.9) where

f_n(α1, γ )=(−1)^n(n−1)/2 Yn

j=1

q^{−(j−1)2γ} θ(q^{αj+···+αn+1})

θ(q^{1+α1−(n+j−2)γ}), (2.10) g_n(ξ, α1, γ )=

Yn j=1

ξ^α_j^{1−2(j−1)γ}θ(q^{α1+1−(n−1)γ}ξj) θ(qξj) · Y

1≤i<j≤n

ξiθ(ξj/ξi)

θ(q^γ+1ξj/ξi). (2.11) Hence,ξj(j ≥1)being replaced byξjq^βin (2.9),

([0, ξ∞]_q : reghηFi)₈0

=(1−q)ⁿ Yn

j=1

(q)³_∞q^{αjγ (j−1)}

θ(q^{αj+···+αn+1})· f_n(α1, γ )·g_n(q^βξ, α1, γ ). (2.12) It is also possible to evaluate explicitlyR

hξFi80(t)·$itself. In fact, the following formula is known (see [16]).

Proposition 2 Z

hξFi80(t)·$

=q^An Yn j=1

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 ) .

In particular, we have (hξFi: reghηFi)₈0

=(1−q)ⁿ(q)³ⁿ_∞·q^Kn· Yn j=1

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), (2.13) where Kndenotes

K_n = −2

3n(n−1)(2n−1)γ²−n(n−1)

2 γ+nβ{α1−(n−1)γ}

+nα1{1+(n−1)γ}. (2.14)

Next, we take as8(t)the q-multiplicative function 8(t)=t₁^α1· · ·t_n^αn (t1· · ·tn)∞

(q^βt1· · ·tn)∞

Y

1≤i<j≤n

(q^1−γtj/ti)∞

(q^γtj/ti)∞ (2.15)

for αj = α1+(j −1)(1−2γ ).Let80(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

Yn j=1

ξ^αj^{1−2(j−1)γ}

θ(qξ1· · ·ξn) θ(q^1+βξ1· · ·ξn)

Y

1≤i<j≤n

θ(qξj/ξi)

θ(q^1+γξj/ξi). (2.16)

3. B_n,C_n,D_n,G₂,F₄and E₆ ∼E₈type root systems

For B_n-type, we take as8(t)the multiplicative function 8(t)=t₁^α1· · ·t_n^αn

Yn j=1

(t_jq^1−β)_∞ (tjq^β)∞

Y

1≤i<j≤n

(q^1−γt_j/t_i)_∞·(q^1−γt_it_j)_∞

(q^γtj/ti)∞·(q^γtitj)∞ (3.1)

forαj =α1+(j −1)(1−2γ ), α1 = ¹₂ −β.This is exactly the one obtained from (1.1) for the root system Bnwhen 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

80(t) = 8(t)A¡

t₁^−(1/2)+nt₂^−(3/2)+n· · ·t_n^(1/2)¢

= (−1)^n(n+1)/28(t)(t₁· · ·t_n)^−n+(1/2) Yn

j=1

(1−t_j)

· Y

1≤i<j≤n

(ti−tj)(1−titj), (3.2)

whereAdenotes the alternating sum with respect to the Weyl group W. X consists of the points x=(ν1, . . . , νn)∈Zⁿ.

Letψ˜_∗(ξ)denote the productξ^λ0Q

α∈R₊ θ(qξ^α)

θ(q^1+γ(α,α)ξ^α) in the RHS of (1.7). Thenψ˜_∗(ξ)is represented more concretely as

ψ˜∗(ξ)= Yn j=1

ξ^αj^j−(j−12)

θ(qξj) θ(q^β+1ξj)

Y

1≤i<j≤n

θ(qξj/ξi)θ(qξiξj)

θ(q^1+γξj/ξi)θ(q^1+γξiξj). (3.3) The explicit formula for C₁has given by Ito [13] in case n=2.

Similarly for Cn-type, we take as8(t)the function 8(t)=t₁^α1· · ·t_n^αn

Yn j=1

¡q^1−βt²_j¢

¡ ∞

q^βt²_j¢

∞

Y

1≤i<j≤n

(q^1−γt_j/t_i)_∞·(q^1−γt_it_j)_∞

(q^γtj/ti)∞·(q^γtitj)∞ (3.4) forα1=1−2β, αj=α1+(j−1)(1−2γ ). This corresponds to (1.1) forγ=γ2, β=γ4. The Weyl group W is isomorphic to the one of Bn-type.80(t)is equal to8(t)A(t₁ⁿt₂ⁿ⁻¹· · ·tn).

X consists of the points x=(ν1, . . . , νn)∈Zⁿor(ν1+¹₂, . . . , νn+¹₂)∈Zⁿ+(¹₂, . . . ,¹₂).

ψ˜_∗(ξ)is then given as ψ˜_∗(ξ)=

Yn j=1

ξ^αj^j−j

θ¡ qξ²j

¢ θ¡

q^β+1ξ²_j¢ Y

1≤i<j≤n

θ(qξj/ξi)θ(qξiξj)

θ(q^1+γξj/ξi)θ(q^1+γξiξj). (3.5) For Dn-type we take as8(t)the function

8(t)= Yn j=1

t^α_j^j−j+1 Y

1≤i<j≤n

(q^1−γtj/ti)∞·(q^1−γtitj)∞

(q^γtj/ti)∞·(q^γtitj)∞ (3.6) forα1 =0 andαj =α1+(j−1)(1−2γ ). This corresponds to (1.1) forγ =γ2. 80(t)is equal to8(t)A(t₁ⁿt₂ⁿ⁻¹· · ·tn).The Weyl group W is isomorphic to the semi-direct product of Zⁿ₂andSn.X is the same as in case of Cntype. ψ˜∗(ξ)is then equal to

ψ˜∗(ξ)= Yn j=1

ξ^αj^j−(j−1)

Y

1≤i<j≤n

θ(qξj/ξi)θ(qξiξj)

θ(q^1+γξj/ξi)θ(q^1+γξiξj). (3.7)

For E8-type we can choose an orthonormal basis{²j}⁸j=1in h^∗∼=R⁸such that the positive simple roots are

ω1=1

2(²1+²8−²2−²3−²4−²5−²6−²7),

ω2=²1+²2, ω3=²2−²1, ω4 =²3−²2, ω5=²4−²3, ω6=²5−²4, ω7=²6−²5, ω8=²7−²6.

The positive roots are±²i+²j(1≤i < j ≤8)and¹₂(²8+P7

i=1ν(i)²i)where{ν(k)}⁷_k=1 moves over the set±1 such that the number ofν(k)which are equal to−1 is even. Then 8(t)has the quasi-symmetry when and only when

αj =(j−1)(1−2γ )(1≤ j ≤7), α8=23(1−2γ ).

8(t)= Y8

j=1

t^α_j^j Y

1≤i<j≤8

(q^1−γtj/ti)∞(q^1−γtitj)∞

(q^γtj/ti)∞(q^γtitj)∞

· Y

ν(k)=±1

¡q^1−γt₁^ν(1)/2· · ·t₇^ν(7)/2t₈^1/2¢

¡ ∞

q^γt₁^ν(1)/2· · ·t₇^ν(7)/2t₈^1/2¢

∞

. (3.8)

The pairing betweenµ=P8

j=1µj²j ∈h^∗and x =(x₁, . . . ,x₈)∈h(∼=R⁸)is given by µ,x→

X8 j=1

µjxj.

X is the eight-dimensional lattice in h consisiting of the points x =(x1, . . . ,x6,x7,x8)

∈h such thatα(x)≡0(Z)forα∈ R₊.In other words, x∈ X consists of the points x such that xj =νj, (1≤ j ≤8)or xj =νj+¹₂, (1≤ j ≤8)satisfying the equality

ν1+ · · · +ν8≡0(2).

ψ˜_∗(ξ)is then equal to ψ˜_∗(ξ)=

( ₇ Y

j=1

ξ⁻_j^2(j−1)γ )

ξ₈^−46γ Y

1≤i<j≤8

θ(qξj/ξi)θ(qξiξj) θ(q^1+γξj/ξi)θ(q^1+γξiξj)

· Y

ν(k)=±1

θ¡

qξ1^ν(1)/2· · ·ξ7^ν(7)/2ξ8^1/2

¢ θ¡

q^γ+1ξ1^ν(1)/2· · ·ξ7^ν(7)/2ξ8^1/2

¢. (3.9)

For E7-type, in terms of the above basis{²j}⁸j=1,the positive simple roots are ω1=1

2(²1+²8−²2−²3−²4−²5−²6−²7),

ω2=²1+²2, ω3=²2−²1, ω4 =²3−²2, ω5=²4−²3, ω6=²5−²4, ω7=²6−²5.

The positive roots are±²i+²j(1≤i < j ≤6),−²7+²8and¹₂(−²7+²8+P6

i=1ν(i)²i) where{ν(k)}⁶_k=1moves over the set±1 such that the number ofν(k)which are equal to−1 is odd. Then8(t)has the quasi-symmetry when and only when

αj =(j−1)(1−2γ )(1≤ j ≤6), −α7=α8= 17

2 (1−2γ ).

8(t)= (Y6

j=1

t^α_j^j )

t₈^α8· ( Y

1≤i<j≤6

(q^1−γt_j/t_i)_∞(q^1−γt_it_j)_∞ (q^γtj/ti)∞(q^γtitj)∞

)

·(q^1−γt₈)_∞ (q^γt8)∞

· Y

ν(k)=±1

¡q^1−γt₁^ν(1)/2· · ·t₆^ν(6)/2t₈^1/2¢

¡ ∞

q^γt₁^ν(1)/2· · ·t₆^ν(6)/2t₈^1/2¢

∞

. (3.10)

X consists of the points x such that either xj=νj(1≤ j≤8),x7=0 whereP₆

j=1νj+ ν8≡0(2),or xj =νj+¹₂(1≤ j ≤6),x7=0,x8=ν8whereP6

j=1νj+ν8 ≡1(2).

We normalizeξasξ7=1. ψ˜_∗(ξ)is then equal to ψ˜_∗(ξ)=

( ₆ Y

j=1

ξ⁻j^2(j−1)γ

)

·ξ8^−17γ

Y

1≤i<j≤6

θ(qξj/ξi)θ(qξiξj)

θ(q^1+γξj/ξi)θ(q^1+γξiξj)· θ(qξ8) θ(q^γ+1ξ8)

· Y

ν(k)=±1

θ¡

qξ₁^ν(1)/2· · ·ξ₆^ν(6)/2ξ₈^1/2¢ θ¡

q^γ+1ξ₁^ν(1)/2· · ·ξ₆^ν(6)/2ξ₈^1/2¢. (3.11) For E6-type, in R⁸the positive simple roots are

ω1=1

2(²1+²8−²2−²3−²4−²5−²6−²7),

ω2=²1+²2, ω3=²2−²1, ω4 =²3−²2, ω5=²4−²3, ω6=²5−²4. The positive roots are±²i+²j(1 ≤ i < j ≤ 5)and ¹₂(²8−²6 −²7+P5

i=1ν(i)²i) where{ν(k)}⁶_k=1moves over the set±1 such that the number ofν(k)which are equal to−1 is even. Then8(t)has the quasi-symmetry when and only when

αj =(j−1)(1−2γ )(1≤ j ≤5), −α6= −α7=α8=4(1−2γ ).

8(t)= ( ₅

Y

j=1

t^α_j^j )

t₈^α8· ( Y

1≤i<j≤5

(q^1−γtj/ti)∞(q^1−γtitj)∞

(q^γtj/ti)_∞(q^γtitj)_∞ )

· Y

ν(k)=±1

¡q^1−γt₁^ν(1)/2· · ·t₅^ν(5)/2t₈^1/2¢

¡ ∞

q^γt₁^ν(1)/2· · ·t₅^ν(5)/2t₈^1/2¢

∞

. (3.12)

X consists of the points x such that either x_j = νj(1 ≤ j ≤ 8),x₆ = x₇ =0 where P5

j=1νj+ν8 ≡0(2),or x_j = νj + ¹₂(1 ≤ j ≤ 5),x₆ = x₇ =0,x₈ =ν8− ¹₂ where P5

j=1νj+ν8≡1(2).