  The paper deals with the analytic theory of the quantum q-deformed Toda
chain; 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.
