LIOUVILLE TYPE MODELS IN THE GROUP-THEORY FRAMEWORK .1. FINITE-DIMENSIONAL ALGEBRAS

In this series of papers we represent the ''Whittaker'' wave functiona l of the (d + 1)-dimensional Liouville model as a correlator in (d + 0 )-dimensional theory of the sine-Gordon type (for d = 0 and 1). The as ymptotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple Gamma function fac tors over all positive roots of the corresponding algebras (finite-dim ensional for d = 0 and affine for d = 1). This is in nice corresponden ce with the recent results on two- and three-point correlators in the 1 + 1 Liouville model, where emergence of peculiar double periodicity is observed. The Whittaker wave functions of (d + 1)-dimensional nonaf fine (''conformal'') Toda type models are given by simple averages in the (d + 0)-dimensional theories of the affine Toda type. This phenome non is in obvious parallel with representation of the free field wave functional, which was originally a Gaussian integral over the interior of a (d + 1)-dimensional disk with given boundary conditions, as a (n onlocal) quadratic integral over the d-dimensional boundary itself. In this paper we concentrate on the finite-dimensional case. The results for finite-dimensional ''Iwasawa'' Whittaker functions are known, and we present a survey. We also construct new ''Gauss'' Whittaker functi ons.