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Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 4, Pages 37–49 (Mi faa2929)  

This article is cited in 4 scientific papers (total in 5 papers)

Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products

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: 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 $\mathbb{R}^*_+=\{r:r>0\}$ 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 $r\to 0$. 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 $\sigma$-finite measure (Lebesgue measure) $\mathcal L$ on the space of distributions.

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


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English version:
Functional Analysis and Its Applications, 2008, 42:4, 279–289

Bibliographic databases:

UDC: 517.5
Received: 11.08.2008

Citation: A. M. Vershik, M. I. Graev, “Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products”, Funktsional. Anal. i Prilozhen., 42:4 (2008), 37–49; Funct. Anal. Appl., 42:4 (2008), 279–289

Citation in format AMSBIB
\by A.~M.~Vershik, M.~I.~Graev
\paper Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products
\jour Funktsional. Anal. i Prilozhen.
\yr 2008
\vol 42
\issue 4
\pages 37--49
\jour Funct. Anal. Appl.
\yr 2008
\vol 42
\issue 4
\pages 279--289

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    This publication is cited in the following articles:
    1. A. M. Vershik, M. I. Graev, “Integral models of representations of the current groups of simple Lie groups”, Russian Math. Surveys, 64:2 (2009), 205–271  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Müller Ch., Neeb K.-H., Seppänen H., “Borel-Weil theory for root graded Banach-Lie groups”, Int. Math. Res. Not. IMRN, 2010, no. 5, 783–823  crossref  mathscinet  zmath  isi
    3. 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
    4. 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
    5. 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
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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