The paper deals with the analytic theory of the quantum q -deformed Toda chains; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N -particle q -deformed open Toda chain are given as a multiple integral of the Mellin-Barnes type. For the periodic chain the two dual Baxter equations are derived. PrefaceIn the late seventies B. Kostant [1] has discovered a fascinating link between the representation theory of non-compact semisimple Lie groups and the quantum Toda chain. Let G be a real split semisimple Lie group, B = MAN its minimal Borel subgroup, let N and V =N be the corresponding opposite unipotent subgroups. Let χ N , χ V be nondegenerate unitary characters of N and V , respectively. Let H T be the space of smooth functions on G which satisfy the functional equationv ∈ V, n ∈ N.

The integral representations for the eigenfunctions of N particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). Both periodic and open N -particle solutions have essentially the same structure being written as a generalized Fourier transform over the eigenfunctions of the N − 1 particle open Toda chain with the kernels satisfying to the Baxter equations of the second and first order respectively. In the latter case this leads to recurrent relations which result to representation of the Mellin-Barnes type for solutions of an open chain. As byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of GL(N, R) group.

The recurrent relations between the eigenfunctions for GL(N, R) and GL(N − 1, R) quantum Toda chains is derived. As a corollary, the Mellin-Barnes integral representation for the eigenfunctions of a quantum open Toda chain is constructed for the

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