The basic representation of the current group O(n,1)X in the L2 space over the generalized Lebesgue measure

Граев, М. И.; Vershik, A. M.

doi:10.1016/s0019-3577(05)80038-2

Cited by 21 publications

(20 citation statements)

References 12 publications

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“…An analog of the measure L θ in the case n > 1 was defined in [17,18] by analogy with the case n = 1. First, we define the vector gamma process with characteristic functional…”

Section: Many Dimensional Generalization Of the Poisson-dirichlet Mea-mentioning

confidence: 99%

Does there exist a lebesgue measure in the infinite-dimensional space?

Vershik

2007

Proc. Steklov Inst. Math.

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I want to write about what I know and remember about the activities of Leonid Vital'evich Kantorovich, an outstanding scientist of the 20th century; about his struggle for recognition of his mathematical economic theories; about the initial stage of the history of linear programming; about the creation of a new area of mathematical activity related to economic applications, which is called sometimes operation research, sometimes mathematical economics, sometimes economic cybernetics, etc.; about its place in the modern mathematical landscape; and, finally, about several personal impressions of this distinguished scientist. My notes in no way pretend to exhaust these topics.

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“…An analog of the measure L θ in the case n > 1 was defined in [17,18] by analogy with the case n = 1. First, we define the vector gamma process with characteristic functional…”

Section: Many Dimensional Generalization Of the Poisson-dirichlet Mea-mentioning

confidence: 99%

Does there exist a lebesgue measure in the infinite-dimensional space?

Vershik

2007

Proc. Steklov Inst. Math.

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“…One of the important applications of nonidentity special representations is as follows: using such a representation, one can construct an irreducible nonlocal unitary representation of the corresponding current group G X , i.e., the group of bounded measurable maps X → G with pointwise multiplication, where X is a measurable space; this question is considered in the series of papers [10,12,14,15,16,17,18,19], [20,21,33]. However, we do not consider these applications in this paper.…”

Section: Introductionmentioning

confidence: 99%

Structure of the complementary series and special representations of the groupsO(n,1) andU(n,1)

Vershik¹,

Граев²

2006

Russ. Math. Surv.

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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

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“…where u ± (g, z) is given by (9). To extend the orthogonal representation U 0 = INT T 0 of P X on the space INT H 0 to the group SL(2, R) X , to each function z ∈ L X we assign the following functional on l 1 + (X):…”

Section: Integral Models Of the Subgroup P ⊂ Sl(2 R) Of Triangular Mmentioning

confidence: 99%

Integral models of unitary representations of current groups with values in semidirect products

Vershik

Граев

2008

Funct Anal Its Appl

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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 P 0 by a one-parameter group R * + = {r : r > 0} of automorphisms of P 0 . This construction is determined by 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 → 0. We apply this construction to the current groups of maximal parabolic subgroups in the groups of motions of the n-dimensional real and complex Lobachevsky spaces. The obtained representations of the current groups of parabolic subgroups uniquely extend to the groups of currents with values in the groups O(n, 1) and U (n, 1). This gives a new description of the representations, constructed in the 1970s and realized in the Fock space, of the current groups of the latter groups. The key role in our construction is played by the so-called special representation of the parabolic subgroup P and a remarkable σ-finite measure (Lebesgue measure) L on the space of distributions.In 1972, on the initiative of I. M. Gelfand, a series of papers by three authors (Israel Moiseevich himself and the present authors) on unitary representations of functional groups, or current groups, was started. The problem that was posed by Gelfand to the first author of this paper in spring 1972 and which 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 is positive, and the first paper [2] on the topic described 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 a technique for working with such representations was developed. Starting from 2004, the authors of the present paper have been carrying out a systematic study of integral models of representations of current groups on the basis of a new interpretation of the continual tensor product; this interpretation 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.

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The basic representation of the current group O(n,1)X in the L2 space over the generalized Lebesgue measure

Cited by 21 publications

References 12 publications

Does there exist a lebesgue measure in the infinite-dimensional space?

Does there exist a lebesgue measure in the infinite-dimensional space?

Structure of the complementary series and special representations of the groupsO(n,1) andU(n,1)

Integral models of unitary representations of current groups with values in semidirect products

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