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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 2(386), Pages 5–72 (Mi umn9260)  

This article is cited in 3 scientific papers (total in 4 papers)

Integral models of representations of the current groups of simple Lie groups

A. M. Vershika, M. I. Graevb

a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b Scientific Research Institute for System Studies of RAS

Abstract: For the class of locally compact groups $P$ that can be written as the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group $\mathbb R^*_+$ of automorphisms of $P_0$, a new model of representations of the current groups $P^X$ is constructed. The construction is applied to the maximal parabolic subgroups of all simple groups of rank 1. In the case of the groups $G=\mathrm{SO}(n,1)$ and $G=\mathrm{SU}(n,1)$, an extension is constructed of representations of the current groups of their maximal parabolic subgroups to representations of the current groups $G^X$. The key role in the construction is played by a certain $\sigma$-finite measure (the infinite-dimensional Lebesgue measure) in the space of distributions.
Bibliography: 32 titles.

Keywords: current group, integral model, Fock representation, canonical representation, special representation, infinite-dimensional Lebesgue measure.

DOI: https://doi.org/10.4213/rm9260

Full text: PDF file (973 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2009, 64:2, 205–271

Bibliographic databases:

UDC: 517.5
MSC: Primary 22E65, 22E46, 22D12; Secondary 58D20
Received: 24.12.2008

Citation: A. M. Vershik, M. I. Graev, “Integral models of representations of the current groups of simple Lie groups”, Uspekhi Mat. Nauk, 64:2(386) (2009), 5–72; Russian Math. Surveys, 64:2 (2009), 205–271

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Vershik, M. I. Graev, “Poisson model of the Fock space and representations of current groups”, St. Petersburg Math. J., 23:3 (2012), 459–510  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. A. M. Vershik, M. I. Graev, “Cohomology in Nonunitary Representations of Semisimple Lie Groups (the Group $U(2,2)$)”, Funct. Anal. Appl., 48:3 (2014), 155–165  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. M. Vershik, M. I. Graev, “Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$”, J. Math. Sci. (N. Y.), 232:2 (2018), 99–120  mathnet  crossref
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