Determinant Formula for Solutions of the Quantum Knizhnik–Zamolodchikov Equation Associated with $U_q(sl_n)$ at $|q| = 1$

We construct the hypergeometric solutions for the quantized Knizhnik-Zamolodchikov equation with values in a tensor product of vector representations of Uq(sln) at \q\ = 1 and give an explicit formula for the corresponding determinant in terms of the double sine function. Introduction In this paper we study the hypergeometric solutions of the quantized Knizhnik-Zamolodchikov (qKZ) equation with values in a tensor product of vector representations of Uq(sln), see Section 1 for the precise formulation of the problem. It is known that the qKZ equation respects the weight decomposition of the tensor product. For each weight subspace we construct a fundamental matrix solution of the qKZ equation and explicitly calculate the corresponding determinant, see Theorem 3.1. Formal integral representations for solutions of the qKZ equation in the sln case, both in the rational and trigonometric situation, were constructed in [TV1]. Though to write down the phase function explicitly in the trigonometric situation it had been assumed in [TV1] that the multiplicative step p of the qKZ equation is inside the unit circle: 0 < \p\ < 1, all the construction in [TV1] used only difference equations for the phase function and after obvious modifications remained valid for an arbitrary step p / 0,1. However, the problem of integrating the formal integral representations suitably and getting in this way Received May 24, 1999. Revised July 21, 1999. 1999 Mathematics Subject Classification(s): 33D80, 17B37, 81R50. 0 Research Fellow of the Japan Society for the Promotion of Science. * Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. ** St. Petersburg Branch of Steklov Mathematical Institute Fontanka 27, St. Petersburg 191011, Russia. 872 TETSUJI MIWA, YosfflHiRO TAKEYAMA AND VITALY TARASOV actual solutions of the qKZ equation is much more analytically involved; one can see this looking at the sh case. In the last four years the hypergeometric solutions of the qKZ equation in the s/2 case were studied quite well. The generic situation was considered in [TV2] (the rational case) and in [TV3] (the trigonometric case for 0 < \p\ < 1). The construction was generalized to the trigonometric case for \p\ = 1 in [MT1] and to the elliptic case of the quantized Knizhnik-Zamolodchikov-Bernard (qKZB) equation in [FTV1], [FTV2]. If some of the representations are finite-dimensional, the situation is no more generic. Rather detailed study of this case has been done in [MV1]; see also [S], [JKMQ], [NPT], [Tl] for some important particular cases. Less is known for the case of n > 2. Some integral formulae for solutions of the qKZ equation were obtained in [S], [KQ], [N], [MT2], but in all the considered cases solutions takes values in a tensor product of vector representations. Recently Varchenko and the third author have managed to extend the construction of [TV2], [TVS] to the higher rank case and get solutions taking values in a tensor product of arbitrary highest weight representations [TV4]. Let us also mention a paper [M], where integral formulae for solutions of another type of the qKZ equation were suggested. In this paper we evaluate a determinant of a certain matrix whose entries are given by multidimensional integrals of ^-hypergeometric type. In the case of ordinary multidimensional hypergeometric integrals a problem of evaluating similar determinants appears, say, in studying arrangements of hyperplanes, and several results have been obtained in this direction, see for instance [VI], [L], [LS], [DT], [MTV], [MV2]. In some particular cases these determinant formulae have another meaning; namely they imply that under certain assumptions the hypergeometric solutions of the differential Knizhnik-Zamolodchikov equation form a basis of solutions [SV], [V2]. There are similar determinant formulae for solutions of the qKZ equation in the sh case. They have been obtained for the rational case in [TV2], [Tl], and for the trigonometric case in [TV3] for 0 < \p\ < 1 and in [MT1] for \p\ = l. It turns out that there is a nice connection of constructions given in [TV3] and [MT1], which in particularly allows to derive the determinant formula for \p\ — 1 from the determinant formula for 0 < \p\ < 1. This subject will be addressed elsewhere [T2]. The paper is organized as follows. The first section contains preliminaries and precise definitions on the qKZ equation. In Section 2 we construct the hypergeometric pairing and give integral formulae for solutions of the qKZ equation. The main result of the paper is formulated in Section 3, see Theorem 3.1. We show that both the left hand side and the right hand side of formula (3.2) satisfy the same system of difference equations and have to be proportional. To compute the proportionality coefficient we study suitable asymptotics of the hypergeometric solutions. We see that the proportionality QUANTUM KZ EQUATION 873 coefficient splits into a product of contributions of each tensor factor, which are calculated in Section 5. In the last Section we complete the proof of Theorem 3.1. A short Appendix contains the necessary information of the double sine function for the convenience of the reader. Acknowledgement V. Tarasov thanks the RIMS, Kyoto University for warm hospitality during his stay there, when most of the results of this paper had been obtained. He also thanks A. Varchenko for valuable remarks. § 1. The Quantized Knizhnik-Zamolodchikov Equation Consider the vector representation V of sln: