Elliptic Quantum Many-body Problem and Double Affine Knizhnik-zamolodchikov Equation Hep-th/9403136 Section 0. Introduction Section 1. Double Hecke Algebras Section 2. Aane R-matrices Section 3. Dunkl Operators and Kz Section 4. Examples 0. Introduction

The elliptic-matrix quantum Olshanetsky-Perelomov problem is introduced for arbitrary root systems by means of an elliptic generalization of the Dunkl operators. Its equivalence with the double aane generalization of the Knizhnik-Zamolodchikov equation (in the induced representations) is established. We generalize the aane the Knizhnik-Zamolodchikov equation from Ch1,2,3] replacing the corresponding root systems by their aane counterparts. To explain the construction in the case of the root system of gl n , let us rst introduce the aane Weyl group S a n. It is the semi-direct product of the symmetric group S n and the lattice A = n?1 i=1 Z ii+1 , where the rst acts on the second permuting f i ; ij = i ? j g naturally. This group is generated by the adjacent transpositions s i = (ii + 1); 1 i < n; and s 0 = s 1] n1 ; where s k] ij = (ij)(k ij) 2 S a n :