The Basic Representation of the Current Group O(n,1)^X in the L^2 space over the generalized Lebesgue Measure

Abstract

We give the realization of the representation of the current group O(n,1)^X where X is a manifold, in the Hilbert space of L^2(F,\nu) of functionals on the the space F of the generalized functions on the manifold X which are square integrable over measure \nu which is related to a distinguish Levy process with values in R^{n-1} which generalized one dimensional gamma process. Unipotent subgroup of the group O(n,1)^X acts as the group of multiplicators. Measure \nu is sigma-finite and invariant under the action current group O(n-1)^X. Ther case of n=2 (SL(2,R^X)) was considered before in the series of papers starting from the article Vershik-Gel'fand-Graev (1973).