  Using the representation theory of $\frak{gl}(N,\RR)$, we express the wave
function of the $GL(N,\RR)$ Toda chain, which two of us recently obtained by
the Quantum Inverse Scattering Method, in terms of multiple integrals. The main
tool is our generalization of the Gelfand-Zetlin method to the case of
infinite-dimensional representations of $\frak{gl}(N,\RR)$. The interpretation
of this generalized construction in terms of the coadjoint orbits is given and
the connection with the Yangian $Y(\frak{gl}(N))$ is discussed. We also give
the hyperbolic Sutherland model eigenfunctions expressed in terms of integrals
in the Gelfand-Zetlin representation. Using the example of the open Toda chain,
we discuss the connection between the Quantum Inverse Scattering Method and
Representation Theory.
