International Congress of Mathematicians

566 Pavel Etingofsatisfies the QDYBE with spectral parameter (for V = W); it is the appropriategeneralization of the exchange operator (see [EFK] for details).The operators R vw (z,X), essentially, represent the monodromy of the KZ differentialequation. In particular, R vw is "almost constant" in u: its matrix elementsin a homogeneous basis under h are powers of e 2nm (in fact, Ry V (u,X) is gaugeequivalent, in an appropriate sense, to a solution of QDYBE without spectral parameter).Similarly, the operator Ry\y(z, A) for q ^ 1 represents the q-monodromy(Birkhoff's connection matrix) of the q-KZ equation. In particular, the matrix elementsof Rvw(u,X) are quasiperiodic in u with period r = \ogp/2ni. Since theyare also periodic with period 1 and meromorphic, they can be expressed rationallyvia elliptic theta-functions.Example 3.6 Let g = sl n , and V the -vector representation of U q (g). If q ^ 1,the exchange operator Ryy(u,X) is a solution of QDYBE, gauge equivalent to thebasic elliptic solution with spectral parameter (see [Mo] and references therein, andalso [FR, EFK]). The gauge transformation involves an exact multiplicative 2-form, which expresses via q-Gamma functions with q = p. Similarly, if q = 1,the exchange operators R vv (u,X) are gauge equivalent to the basic trigonometricsolution without spectral parameter, with q = e^/">-(k+g). the gauge transformationinvolves an exact multiplicative 2-form expressing via classical Gamma-functions.This is obtained by sending q to 1 in the result of [Mo].Remark Note that the limit q —¥ 1 in this setting is rather subtle. Indeed,for q ^ 1 the function Jy\y(u,X) has an infinite sequence of poles in the z-plane,which becomes denser as q approaches 1 and eventually degenerates into a branchcut; i.e. this single valued meromorphic function becomes multivalued in the limit.3.11. The quantum Kazhdan-Lusztig functorLet g = sl n , q ^ 1. Example 3.6 allows one to construct a tensor functorfrom the category fiepf(U q (gj) of finite dimensional C/ g (g)-modules, to the categoryRepj(72) of finite dimensional representations of the basic elliptic solution72 of QDYBE with spectral parameter (i.e. to the category of finite dimensionalrepresentations of Felder's elliptic quantum group). Namely, let V be the vectorrepresentation, and for any finite dimensional representation W of U q (g), setLw = Rvw- Then (W,Lw) is a representation of the dynamical R-matrix Ryy.This defines a functor F : Repf(U q (g)) —¥ Repf(Ryy). Moreover, this functor isa tensor functor: the tensor structure F(W) ® F(U) —¥ F(W ® U) is given by thefusion operator Jwu(X) (the axiom of a tensor structure follows from the dynamicaltwist equation for J). On the other hand, since Ryy and 72 are gauge equivalentby an exact form, their representation categories are equivalent, so one may assumethat F lands in Rep^(72).If the scalars for Repf(U q (g)) are taken to be the field of periodic functionsof A (in particular, fc is regarded as a variable), then F is fully faithful; it can beregarded as a q-analogue of the Kazhdan-Lusztig functor (see [EM] and referencestherein). It generalizes to infinite dimensional representations, and allows one to