The Basic Representation of the Current Group O(n,1)ˣ in the L² space over the generalized Lebesgue Measure

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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 L²(F,ν) of functionals on the the space F of the generalized functions on the manifold X which are square integrable over measure νwhich is related to a distinguish Levy process with values in Rⁿ⁻¹ which generalized one dimensional gamma process. Unipotent subgroup of the group O(n,1)ˣ acts as the group of multiplicators. Measure νis sigma-finite and invariant under the action current group O(n-1)ˣ. Ther case of n=2 (SL(2,Rˣ)) was considered before in the series of papers starting from the article Vershik-Gel'fand-Graev (1973).