Central Elements for Quantum Affine Algebras and Affine Macdonald’s Operators

We describe a generalization of Drinfeld’s description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald’s difference operators. 1. The center of Uq(g), where g is a simple Lie algebra. 1.1. Let g be a simple Lie algebra over C of rank r. Let h be a Cartan subalgebra in g. Let W be the Weyl group. We fix a Weyl group invariant inner product 〈, 〉 on h and h by setting 〈α, α〉 = 2 for short roots. Fix a polarization of g. Let ρ ∈ h be the half-sum of positive roots. Let Q be the root lattice of g, Q be the semigroup with 0 spanned by the positive roots. Let α1, ..., αr be the simple roots. Let A = (aij), aij = 2〈αi, αj〉/〈αi, αi〉 be the Cartan matrix of g. Let P be the weight lattice of g, P be the set of dominant integral weights, and ω1, ..., ωr be the fundamental weights. Let N be the order of P/Q. Note that for any λ, μ ∈ P N〈λ, μ〉 ∈ Z. 1.2. Let q̂ be a formal variable. Let q = q̂ . For any a ∈ 1 NZ we define q a := q̂. Let Uq(g) be the quantum group corresponding to g ([Dr1,J1]). It is a Hopf algebra generated over the field F = C(q̂) by the elements Ei, Fi, 1 ≤ i ≤ r, and