Elliptic extension of Gustafson's q-integral of type G2☆
Introduction
The Askey–Wilson integral is a complex integral given by where and is the unit circle traversed in the positive direction. Hereafter, for a fixed satisfying , we use the symbol and the abbreviation . 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 . 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 by van Diejen and Spiridonov [15], Spiridonov [12], Rains [11]. They include the elliptic extension of (1.1) where , under the balancing condition . Here, for fixed p, satisfying , , we denote by the Ruijsenaars elliptic gamma function defined by We also use the notation . Note that satisfies where is a theta function satisfying , and also satisfies The Askey–Wilson integral (1.1) is obtained from (1.2) as a special case, first by replacing with and by taking the limit and 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 (of type I) due to Gustafson [3, p. 101, Theorem 8.1] and [2]. Proposition 1.1 Gustafson Suppose that satisfy . Then we have where and is the 2-dimensional torus given by
Our main result is Theorem 1.2 Suppose that and satisfy . Under the balancing condition , we have
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 with , and then take the limit .
Remark 2 By (1.5), under the condition , where , the right-hand side of (1.7) is also expressed as This coincides with the expression in the conjecture [13, p. 213, (36)] when . Moreover, by the property , the right-hand side of (1.7) is rewritten into
Remark 3 By the constraint , the integrand of the left-hand side of (1.7) is also expressed as This expression consists of two parts, one depending on the short roots and the other on the long roots . In the proof of (1.7) we will use the coordinates associated with the simple roots, defined as
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 case including the formula (1.2), which might be simpler than the case of this paper.
This paper is organized as follows. After defining basic terminology of the root system 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 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 , where and 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 , of the operator . 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 . 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 naturally arises in the process of calculation of the asymptotic behavior.
Lastly, we comment on our calculation of . For theta functions we need to expand as a linear combination of theta functions which belong to a particular basis of . In this paper we made use of the basis of that consists of the Lagrange interpolation functions associated with the specific points , , and defined in Section 6. (We constructed this basis in a heuristic way. See the set of theta functions , whose interpolation property is presented in the table below (6.16).) In the cases of and 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 . It would be an interesting problem to find a universal way which produces adequate interpolation bases for general root systems.
Section snippets
Root system
Let be the standard basis of with the inner product satisfying , and let V be the hyperplane in with equation , i.e., . Let be the root system of type given by where . We refer the setting of the root system of type to Macdonald's book [10]. We fix the set of simple roots given by The set of
elliptic Gustafson integral and its q-difference equations
Let be function in defined by where and with complex parameters . We also use the notation instead of when we need to make the dependence on the parameters explicit. Through (1.11), coincides with (1.9) or (1.10). For the function we investigate the double integral
Analytic continuation
The integral , regarded as a holomorphic function in , can be continued to a meromorphic function on . We prove this fact by means the q-difference equations (3.3).
In view of Proposition 3.1 we consider the meromorphic function which is also written as (1.8) if , as is mentioned in the introduction. Then it turns out that satisfies the same q
Coboundary operator
In this section we explain a fundamental method for deriving q-difference equations of the contour integrals (3.6) based on an operator . 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 we have where
Proof of Lemma 3.2
The goal of this section is to give a proof of Lemma 3.2 investigating the -linear mapping defined in the previous section. For that purpose we first clarify the structure of the target space in Definition 5.2. Lemma 6.1 . Proof For arbitrary , since is a holomorphic function of , can be expanded as Laurent series , where . Let D be the set defined by , 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 for abbreviation.
As before, we assume that the parameters satisfy the balancing condition , and regard , where , as a function of . By Theorem 4.2 we showed that the meromorphic functions and are related by the formula provided that 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.