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In the series of papers we represent the ``Whittaker'' wave functional of$d+1$-dimensional Liouville model as a correlator in $d+0$-dimensional theoryof the sine-Gordon type (for $d=0$ and $1$). Asypmtotics of this wave functionis characterized by the Harish-Chandra function, which is shown to be a productof simple $\Gamma$-function factors over all positive roots of thecorresponding algebras (finite-dimensional for $d=0$ and affine for $d=1$).This is in nice correspondence with the recent results on 2- and 3-pointcorrelators in $1+1$ Liouville model, where emergence of peculiardouble-periodicity is observed. The Whittaker wave functions of$d+1$-dimensional non-affine ("conformal") Toda type models are given by simpleaverages in the $d+0$ dimensional theories of the affine Toda type. Thisphenomenon is in obvious parallel with representation of the free-field wavefunctional, which is originally a Gaussian integral over interior of a$d+1$-dimensional disk with given boundary conditions, as a (non-local)quadratic integral over the $d$-dimensional boundary itself. In the presentpaper we mostly concentrate on the finite-dimensional case. The results forfinite-dimensional "Iwasawa" Whittaker functions were known, and we presenttheir survey. We also construct new "Gauss" Whittaker functions.Comment: 47 pages, LaTe

Liouville Type Models in the Group Theory Framework: I. Finite-Dimensional Algebras