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The purpose of this paper is to give a construction of representations of the group of currents for semisimple groups of rank greater than one. Such groups have no unitary representations in the Fock space, since the semisimple groups of this form have no nontrivial cohomology in faithful irreducible representations. Thus we first construct cohomology of the semisimple groups in nonunitary representations. The principal method is to reduce all constructions to Iwasawa subgroups (solvable subgroups of the semisimple groups), with subsequent extension to the original group. The resulting representation is realized in the so-called quasi-Poisson Hilbert space associated with natural measures on infinite-dimensional spaces. Key words: Iwasawa subgroup, cohomology, group of currents, nonunitary representations., Comment: 29 pp. Ref.25

We suggest a method of constructing special nonunitary representations of semisimple Lie groups using representations of Iwasawa subgroups. As a typical example, we study the group $U(2,2)$., Comment: 19 pp, Ref 25

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups., Comment: 15 pp. Ref 12; Moscow Math.Journ. v.12, No 3, 2012

This note is an attempt to give an answer for the following old I.M. Gelfand's question: why some important problems of integral geometry (e.g., the Radon transform and others) are related to harmonic analysis on groups, but for other quite similar problems such relations are not clear? In the note we examine standard problems of integral geometry generating harmonic analysis (the Plancherel theorem etc.) on pairs of commutative hypergroups that are in a duality of Pontryagin's type. As a result new meaningful examples of hypergroups are constructed., Comment: 10 pages, to be published in Doklady Mathematics, 2011

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 $P_0$ and a one-parameter group ${\mathbb R {}}^*_+=\{r:r>0\}$ of automorphisms of $P_0$. This construction is determined by a a faithful unitary representation of $P_0$ (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as $r\to 0$. 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 $\sigma$-finite measure (Lebesgue measure) $\mathcal L$ in the space of distributions., Comment: 13 pp, Ref.18

We give a survey of several models of irreducible complementary series representations and their limits, special representations, for the groups SU(n,1) and SO(n,1), including new ones. These groups, whose geometrical meaning is well known, exhaust the list of simple Lie groups for which the identity representation is not isolated in the space of irreducible unitary representations (i.e., which do not have the Kazhdan property) and hence there exist irreducible unitary representations of these groups -- so-called ``special representations'' -- for which the first cohomology of the group with coefficients in these representations is nontrivial. By technical reasons, it is more convenient to consider the groups O(n,1) and U(n,1). Most part of the paper is devoted to the group U(n,1). The main emphasis is on the so-called commutative models of special and complementary series representations: in these models, the maximal unipotent subgroup is represented by multiplicators in the case of O(n,1), and by the canonical model of the Heisenberg representations in the case of U(n,1). Earlier, these models were studied only for the group $SL(2,\RR)$. They are especially important for realization of nonlocal representations of current groups, which will be considered elsewhere. We substantially use the ``density'' of the irreducible representations under study of SO(n,1): their restrictions to the maximal parabolic subgroup $ {{P}} $ are equivalent irreducible representations. Conversely, in order to extend an irreducible representation of ${{P}}$ to a representation of SO(n,1), we must additionally define only one involution. For the group U(n,1), the situation is similar but slightly more complicated., Comment: 80 pp, 54 References

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)., Comment: 26 p. Refs 19