International Congress of Mathematicians

On the Dynamical Yang-Baxter Equation 567construct elliptic deformations of all evaluation representations of U q (gj) (which wasdone for finite dimensional representations in [TV]). The versions of this functorwithout spectral parameter, from representations of g (U q (gj) to representations ofthe basic rational (trigonometric) solution of QDYBE can be found in [EV3].4. Traces of intertwining operators and MacdonaldfunctionsIn this section we discuss a connection between dynamical R-matrices and certainintegrable systems and special functions (in particular, Macdonald functions).4.1. Trace functionsLet V be a finite dimensional representation of U q (g) (q ^ 1), such thatV[0] 7^ 0. Recall that for any v £ V[0] and generic p, one can define an intertwiningoperator $ ß such that ($ ß ) = v. Following [EV4] set ^v(X,p)=Tr|M f ,( < £Jlg 2A )- This is an infinite series in the variables q^( x > ai i (where a, are thesimple roots) whose coefficients are rational functions of qb À > ai ï (times a commonfactor g 2( A^). For generic p this series converges near 0, and its matrix elementsbelong to q 2(x^(C(q (x ' a^) ® C(q (^ai^)).Yet \Py(A, p) be the End(V[0])-valued function, such that ^y(X, pjv = ^V(X,p).The function \Py has remarkable properties and in a special case is closely relatedto Macdonald functions. To formulate the properties of vPy, we will consider arenormalized version of this function. Namely, let 5 q (X) be the Weyl denominatorria>o( < 3' (A ' a)^w acting on functions on h* with values in V[0]. Namely, we set (T>\yfi)(X) =Si/eb* Tr|i4/(72i4/y(^A — pj)f(X + v). These operators are dynamical analogs oftransfer matrices, and were introduced in [FV2]. It can be shown that ÎVi®w 2=T>W 1 'D\Y 2 ; in particular, V\y commute with each other, and the algebra generatedby them is the polynomial algebra in V\y i , where W, are the fundamental representationsof U q (g).4.3. Difference equations for the trace functionsIt turns out that trace functions Fy(X,p), regarded as functions of A, arecommon eigenfunctions of T>\y.