Inverse Harish - Chandra transform and difference operators

In the paper we calculate the images of the operators of multiplication by Laurent polynomials with respect to the Harish-Chandra transform and its non-symmetric generalization due to Opdam. It readily leads to a new simple proof of the Harish-Chandra inversion theorem in the zonal case (see [HC,He1]) and the corresponding theorem from [O1]. We assume that k > 0 and restrict ourselves to compactly supported functions, borrowing the growth estimates from [O1]. The Harish-Chandra transform is the integration of symmetric functions on the maximal real split torus A of a semi-simple Lie group G multiplied by the spherical zonal function φ(X,λ), where X ∈ A, λ ∈ (LieA ⊗R C) . The measure is the restriction of the invariant measure on G to the space of double cosets K\G/K ⊂ A/W for the maximal compact subgroup K ⊂ G and the restricted Weyl group W . The function φ is a symmetric (W -invariant) eigenfunction of the radial parts of the G-invariant differential operators on G/K; λ determines the set of eigenvalues. The parameter k is given by the root multiplicities (k = 1 in the group case). There is a generalization to arbitrary k due to Calogero, Sutherland, Koornwinder, Moser, Olshanetsky, Perelomov, Heckman, and Opdam. See [HO1,H1,O2] for a systematic theory. In the non-symmetric variant due to Opdam [O1], the operators from [C5] replace the radial parts of G-invariant operators and their k-generalizations. The problem is to define the inverse transforms for various classes of functions. In the papers [C1-C4], a difference counterpart of the Harish-Chandra transform was suggested, which is also a deformation of the Fourier transform in the p-adic theory of spherical functions. Its kernel (a q-generalization of φ) is defined as an eigenfunction of the q-difference “radial parts” (Macdonald’s operators and their generalizations). There are applications in combinatorics (the Macdonald polynomials), the representation theory (say, quantum groups at roots of unity), and the mathematical physics (the Knizhnik-Zamolodchikov equations and more). The difference Fourier transform is self-dual, i.e. the