Duality for Knizhinik-Zamolodchikov and Dynamical Equations, and Hypergeometric Integrals

The Knizhnik-Zamolodchikov (KZ ) equations is a holonomic system of differential equations for correlation functions in conformal field theory on the sphere [KZ]. The KZ equations play an important role in representation theory of affine Lie algebras and quantum groups, see for example [EFK]. There are rational, trigonometric and elliptic versions of KZ equations, depending on what kind of coefficient functions the equations have. In this paper we will consider only the rational and trigonometric versions of the KZ equations. The rational KZ equations associated with a reductive Lie algebra g is a system of equations for a function u(z1, . . . , zn) of complex variables z1, . . . , zn , which takes values in a tensor product V1⊗ . . .⊗Vn of g-modules V1, . . . , Vn . The equations depend on a complex parameter κ , and their coefficients are expressed in terms of the symmetric tensor Ω ∈ U(g)⊗ U(g) corresponding to a nondegenerate invariant bilinear form on g . For example, if g = sl2 and e, f, h are its standard generators such that [e,f ] = h , then Ω = e⊗f + f ⊗ e+ h⊗ h/2 . The rational KZ equations are