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We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup P0 and a one-parameter group R ∗ + = {r : r > 0} of automorphisms of P0. This construction is determined by a a faithful unitary representation of P0 (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as r → 0. We apply this construction to the groups of currents of the maximal parabolic subgroups of the groups of motions of the n-dimensional real and complex Lobachevsky spaces. The obtained representations of the groups of parabolic currents can be uniquely extended to the groups of currents with values in the semisimple groups O(n, 1) and U(n, 1). This gives a new description of the representations of the groups of currents of these groups constructed in the 70s and realized in the Fock space. The key role in our construction is played by the so-called special representation of the parabolic subgroup P and the remarkable σ-finite measure (Lebesgue measure) L in the space of distributions. To dear Israel Moiseevich Gelfand on the occasion of his 95th birthday In 1972, on the initiative of I. M. Gelfand, a series of papers by three authors (I.M., M.I., A.M) on unitary representations of functional groups, or current groups, was started. The problem that was posed by I.M. to the first author of this paper in spring 1972 and that initiated this series of papers was to find out whether there exists a “multiplicative integral of representations” (see below) for the group SL(2, R). Soon it became clear that the answer to this question is positive, and the first paper of the series [2] contained a description of several models of this representation. The main idea was to study a neighborhood of the identity representation of the group SL(2, R) itself and the so-called special (infinitesimal) representation of this group, whose first cohomology is nontrivial. A multiplicative integral of representations is, in the authors’ words, a “tensor product of infinitely many infinitesimal representations.” In the subsequent joint papers, various generalizations of this construction (to simple Lie groups of rank 1, for groups of diffeomorphisms, etc.) were found, and techniques for working with such representations were developed. Starting from 2004, the authors of this paper initiated a systematic study of integral models of representations of current groups based on a new interpretation of continual tensor product that is different from the Fock one and essentially exploits a remarkable σ-finite measure in the space of distributions. A systematic presentation of the whole area will be given in the book The Representation Theory of Current Groups by the three authors, which is now in preparation. Supported by the grants NSh-2460.2008.1 and RFBR 08-01-00379. Supported by the RFBR grant 07-01-00101a. Date: August 9, 2008.

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