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

A. M. Vershik·M. I. Graev
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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,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 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 νis 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).

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A. M. Vershik, M. I. Graev. 2005-03-20. The Basic Representation of the Current Group O(n,1)^X in the L^2 space over the generalized Lebesgue Measure. https://arxiv.org/abs/math/0503404

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