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Advances in Mathematics

Volume 370, 26 August 2020, 107211
Advances in Mathematics

Elliptic extension of Gustafson's q-integral of type G2

https://doi.org/10.1016/j.aim.2020.107211Get rights and content

Abstract

The evaluation formula for an elliptic beta integral of type G2 is proved. The integral is expressed by a product of Ruijsenaars' elliptic gamma functions, and the formula includes that of Gustafson's q-beta integral of type G2 as a special limiting case as p0. The elliptic beta integral of type BC1 by van Diejen and Spiridonov is effectively used in the proof of the evaluation formula.

Introduction

The Askey–Wilson integral is a complex integral given by(q;q)2(2π1)T(x2,x2;q)k=14(akx,akx1;q)dxx=(a1a2a3a4;q)1i<j4(aiaj;q), where |ak|<1 (k=1,,4) and T is the unit circle {xC||x|=1} traversed in the positive direction. Hereafter, for a fixed qC satisfying |q|<1, we use the symbol (u;q)=ν=1(1qνu) and the abbreviation (u1,,um;q)=(u1;q)(um;q). The infinite product on the right-hand side of (1.1) is expressed by a product of q-gamma functions. In this sense formula (1.1) can be regarded as a kind of beta integral, and in fact plays a fundamental role in the theory of Askey–Wilson q-orthogonal polynomials [1]. This type of q-beta integrals has been extended to multiple q-beta integrals in the framework of Macdonald theory of multivariable q-orthogonal polynomials associated with root systems. In this context the Askey–Wilson integral (1.1) is of type BC1. In a series of pioneering works around 1990, Gustafson discovered various evaluation formulas for multiple q-beta integrals associated with root systems, including several remarkable identities which are not covered by the so-called Macdonald constant terms. In the cases of non-simply laced root systems, there are basically two types in Gustafson's multiple q-beta integrals, which are later called type I and type II in the context of [15]. (See also [4] for their explicit forms.)

In the last two decades, several elliptic extensions of the q-beta integrals have been studied, especially for those of type BCn by van Diejen and Spiridonov [15], Spiridonov [12], Rains [11]. They include the elliptic extension of (1.1)(p;p)(q;q)2(2π1)Tk=16Γ(akx,akx1;p,q)Γ(x2,x2;p,q)dxx=1i<j6Γ(aiaj;p,q), where |ak|<1 (k=1,,6), under the balancing condition a1a6=pq. Here, for fixed p, qC satisfying |p|<1, |q|<1, we denote by Γ(u;p,q) (uC) the Ruijsenaars elliptic gamma function defined byΓ(u;p,q)=(pqu1;p,q)(u;p,q),where(u;p,q)=μ,ν=0(1pμqνu). We also use the notation Γ(u1,,um;p,q)=Γ(u1;p,q)Γ(um;p,q). Note that Γ(u;p,q) satisfiesΓ(qu;p,q)=θ(u;p)Γ(u;p,q)andΓ(pu;p,q)=θ(u;q)Γ(u;p,q), where θ(u;p)=(u,p/u;p) is a theta function satisfying θ(pu;p)=θ(u;p)/u, and also satisfiesΓ(pqu1;p,q)=1Γ(u;p,q),1Γ(u,u1;p,q)=u1θ(u;p)θ(u;q). The Askey–Wilson integral (1.1) is obtained from (1.2) as a special case, first by replacing a6 with pa6 and by taking the limit p0 and a50 consecutively.

Compared with the development in the cases of classical root systems, the elliptic extensions of the cases of exceptional root systems are not fully studied yet. The aim of this paper is to prove an elliptic extension of the following q-integral formula of type G2 (of type I) due to Gustafson [3, p. 101, Theorem 8.1] and [2].

Proposition 1.1 Gustafson

Suppose that akC(1k4) satisfy |ak|<1. Then we have(q;q)212(2π1)2T21i<j3(xixj,xi1xj,xixj1,xi1xj1;q)i=13k=14(akxi,akxi1;q)dx1x1dx2x2=(a12a22a32a42;q)(a1a2a3a4;q)i=14(ai;q)(ai2;q)1i<j41(aiaj;q)1i<j<k41(aiajak;q), where x3=x11x21 and T2 is the 2-dimensional torus given byT2={(x1,x2)(C)2||xi|=1(i=1,2)}.

Our main result is

Theorem 1.2

Suppose that x1x2x3=1 and akC(1k5) satisfy |ak|<1. Under the balancing condition (a1a2a3a4a5)2=pq, we have(p;p)2(q;q)212(2π1)2T2i=13k=15Γ(akxi,akxi1;p,q)1i<j3Γ(xixj,xi1xj,xixj1,xi1xj1;p,q)dx1x1dx2x2=i=15Γ(ai2;p,q)Γ(ai;p,q)1i<j5Γ(aiaj;p,q)1i<j<k5Γ(aiajak;p,q)1i<j<k<l5Γ(aiajakal;p,q).

In 2007 this formula was communicated as a conjecture by one of the authors (M. Ito) to V. P. Spiridonov. This conjecture was formulated by Spiridonov and Vartanov [14], [13] in the context of duality of superconformal indices. As far as we know, however, no proof of this formula has been given so far.

Remark 1

Gustafson's formula (1.6) is included in (1.7) as a limiting case; first replace a5 with p12a5, and then take the limit p0.

Remark 2

By (1.5), under the condition a1a2a3a4a5=ϵp12q12, where ϵ{1,1}, the right-hand side of (1.7) is also expressed asi=15Γ(ai2;p,q)Γ(ai;p,q)Γ(ϵp12q12ai;p,q)1i<j5Γ(aiaj;p,q)Γ(ϵp12q12aiaj;p,q). This coincides with the expression in the conjecture [13, p. 213, (36)] when ϵ=1. Moreover, by the property Γ(u2;p,q)=Γ(u,u,p12u,p12u,q12u,q12u,p12q12u,p12q12u;p,q), the right-hand side of (1.7) is rewritten intoi=15Γ(ai,p12ai,p12ai,q12ai,q12ai,ϵp12q12ai;p,q)×1i<j5Γ(aiaj;p,q)1i<j<k5Γ(aiajak;p,q).

Remark 3

By the constraint x1x2x3=1, the integrandΦ(x)=i=13k=15Γ(akxi,akxi1;p,q)1i<j3Γ(xixj,xi1xj,xixj1,xi1xj1;p,q) of the left-hand side of (1.7) is also expressed asΦ(x)=i=13k=15Γ(akxi,akxi1;p,q)Γ(xi,xi1;p,q)1j<k31Γ(xjxk1,xkxj1;p,q). This expression consists of two parts, one depending on the short roots {xi,xi1|1i3} and the other on the long roots {xjxk1,xkxj1|1j<k3}. In the proof of (1.7) we will use the coordinates (z1,z2) associated with the simple roots, defined asz1=x1/x2,z2=x2.

Theorem 1.2 will be proved in two steps. The first step is to show that both sides of (1.7) satisfy a common system of q-difference equations, so that we can consequently confirm both sides coincide up to a constant. The second step is to analyze asymptotic behaviors of both sides at a singularity in order to determine the constant. This method is also applicable to other elliptic beta integrals. In particular, we refer to [7] for the BCn case including the formula (1.2), which might be simpler than the G2 case of this paper.

This paper is organized as follows. After defining basic terminology of the root system G2 in Section 2, we first present in Section 3 the explicit forms of the q-difference equations which the integral (1.7) satisfies (Proposition 3.1). In Section 4 we study the analytic continuation of the integral (1.7) as a meromorphic function of the parameters a1,,a4 in a specific domain. We use this argument to show that the integral (1.7) is expressed as a product of elliptic gamma functions up to a constant. In Section 5 we explain a fundamental method, which corresponds to integration by parts in calculus, to deduce the q-difference equations for the contour integral (1.7). This method is formulated in terms of a q-difference coboundary operator sym:GϵF, where Gϵ and F are defined as spaces of theta functions specified by individual quasi-periodicities. Section 6 is a technical part; we investigate in detail the source and target spaces Gϵ, F of the operator sym. We apply this argument to proving Lemma 3.2, which we used to derive the q-difference equations in Proposition 3.1. Section 7 is devoted to asymptotic analysis of the contour integral (1.7) along the singularity a1a2=1. It is used to determine the explicit value of the constant, which was indefinite at the stage of Section 4. It should be noted that the elliptic beta integral (1.2) of type BC1 naturally arises in the process of calculation of the asymptotic behavior.

Lastly, we comment on our calculation of sym. For theta functions φGϵ we need to expand symφF as a linear combination of theta functions which belong to a particular basis of F. In this paper we made use of the basis of F that consists of the Lagrange interpolation functions associated with the specific points p23, p13, p12 and p12(C)2 defined in Section 6. (We constructed this basis in a heuristic way. See the set of theta functions {F1(z),F2(z),F3(z),G(z)}, whose interpolation property is presented in the table below (6.16).) In the cases of An and BCn root systems in [8] and [9], [6], [7], [5], respectively, we remark that Lagrange interpolation functions in a space of theta functions of particular quasi-periodicity are introduced by systematically specifying a set of reference points in (C)n. It would be an interesting problem to find a universal way which produces adequate interpolation bases for general root systems.

Section snippets

Root system G2

Let {ε1,ε2,ε3} be the standard basis of R3 with the inner product (,) satisfying (εi,εj)=δij, and let V be the hyperplane in R3 with equation ξ1+ξ2+ξ3=0, i.e., V={ξR3|(ξ,ε1+ε2+ε3)=0}. Let RV be the root system of type G2 given byR={±ε¯1,±ε¯2,±ε¯3}{±(ε¯1ε¯2),±(ε¯1ε¯3),±(ε¯2ε¯3)}, where ε¯i=εi(ε1+ε2+ε3)/3. We refer the setting of the root system of type G2 to Macdonald's book [10]. We fix the set of simple roots {α1,α2}R given byα1=ε¯1ε¯2=ε1ε2,α2=ε¯2=(ε1+2ε2ε3)/3. The set of

G2 elliptic Gustafson integral and its q-difference equations

Let Φ(z) be function in z=(z1,z2)(C)2 defined byΦ(z)=Φ+(z)Φ+(z1), where z1=(z11,z21) andΦ+(z)=k=15Γ(akz2,akz1z2,akz1z22;p,q)Γ(z2,z1z2,z1z22,z1,z1z23,z12z23;p,q) with complex parameters a=(a1,,a5)C. We also use the notation Φ(a;z)=Φ(a1,,a5;z) instead of Φ(z) when we need to make the dependence on the parameters a=(a1,,a5) explicit. Through (1.11), Φ(z) coincides with (1.9) or (1.10). For the function Φ(z) we investigate the double integralI=σΦ(z)ϖ(z),ϖ(z)=ϖ(z1,z2)=1(2π1)2dz1z1dz2z2

Analytic continuation

The integral I(a1,,a5), regarded as a holomorphic function in (a1,,a4)U0, can be continued to a meromorphic function on (C)4. We prove this fact by means the q-difference equations (3.3).

In view of Proposition 3.1 we consider the meromorphic functionJ(a1,,a5)=i=15Γ(ai2;p,q)Γ(ai;p,q)1i<j5Γ(aiaj;p,q)1i<j<k5Γ(aiajak;p,q)×1i<j<k<l5Γ(aiajakal;p,q), which is also written as (1.8) if (a1a5)2=pq, as is mentioned in the introduction. Then it turns out that J(a1,,a5) satisfies the same q

Coboundary operator sym

In this section we explain a fundamental method for deriving q-difference equations of the contour integrals (3.6) based on an operator sym. This method corresponds to integration by parts in calculus, and will be used in the succeeding section for the proof of Lemma 3.2 presented in Section 3.

From the definition (3.1) of Φ(z) we haveTq,z1Φ(z)Φ(z)=f+(z)Tq,z1f(z), wheref+(z)=z11z232k=15θ(akz1z2,akz1z22;p)θ(z1z2,z1z22,z1,z1z23,z12z23;p),f(z)=f+(z1)=z1z232k=15θ(akz11z21,akz11z22;p)θ(z1

Proof of Lemma 3.2

The goal of this section is to give a proof of Lemma 3.2 investigating the C-linear mapping sym:GϵF defined in the previous section. For that purpose we first clarify the structure of the target space F in Definition 5.2.

Lemma 6.1

dimCF4.

Proof

For arbitrary f(z)F, since f(z) is a holomorphic function of z=(z1,z2)(C)2, f(z) can be expanded as Laurent series λPcλzλ, where λ=λ1α1+λ2α2Q=P. Let D be the set defined by D={λP|(λ,α1)0,(λ,α2)0,(λϖ1,ϖ1)0}, which is the set of points in the triangle area

Computation of the constant b

In this section we use the double-sign symbol like Γ(uv±1;p,q)=Γ(uv,uv1;p,q) for abbreviation.

As before, we assume that the parameters satisfy the balancing condition (a1a5)2=pq, and regard a5=ϵp12q12/a1a4, where ϵ{1,1}, as a function of (a1,,a4). By Theorem 4.2 we showed that the meromorphic functions I(a1,,a5) and J(a1,,a5) are related by the formulaI(a1,,a5)=bJ(a1,,a5) provided that |p| is sufficiently small. To determine the constant b, we investigate the behavior of these two

Acknowledgments

The authors are grateful to the anonymous referee for valuable comments that have helped them improve the manuscript.

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This work is supported by JSPS Kakenhi Grants (B)15H03626 and (C)18K03339.

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