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Uspekhi Mat. Nauk, 1973, Volume 28, Issue 5(173), Pages 83–128 (Mi umn4953)  

This article is cited in 49 scientific papers (total in 51 papers)

Representations of the group $SL(2,\mathbf R)$, where $\mathbf R$ is a ring of functions

A. M. Vershik, I. M. Gel'fand, M. I. Graev


Abstract: We obtain a construction of the irreducible unitary representations of the group of continuous transformations $X\to G$, where $X$ is a compact space with a measure $m$ and $G=PSL(2,\mathbf R)$, that commute with transformations in $X$ preserving $m$. This construction is the starting point for a non-commutative theory of generalized functions (distributions). On the other hand, this approach makes it possible to treat the representations of the group of currents investigated by Streater, Araki, Parthasarathy, and Schmidt from a single point of view.

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English version:
Russian Mathematical Surveys, 1973, 28:5, 87–132

Bibliographic databases:

UDC: 517.5
MSC: 22D10, 17B15
Received: 15.06.1973

Citation: A. M. Vershik, I. M. Gel'fand, M. I. Graev, “Representations of the group $SL(2,\mathbf R)$, where $\mathbf R$ is a ring of functions”, Uspekhi Mat. Nauk, 28:5(173) (1973), 83–128; Russian Math. Surveys, 28:5 (1973), 87–132

Citation in format AMSBIB
\Bibitem{VerGelGra73}
\by A.~M.~Vershik, I.~M.~Gel'fand, M.~I.~Graev
\paper Representations of the group $SL(2,\mathbf R)$, where $\mathbf R$ is a~ring of functions
\jour Uspekhi Mat. Nauk
\yr 1973
\vol 28
\issue 5(173)
\pages 83--128
\mathnet{http://mi.mathnet.ru/umn4953}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=393337}
\zmath{https://zbmath.org/?q=an:0297.22003}
\transl
\jour Russian Math. Surveys
\yr 1973
\vol 28
\issue 5
\pages 87--132
\crossref{https://doi.org/10.1070/RM1973v028n05ABEH001616}


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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, I. M. Gel'fand, M. I. Graev, “Representations of the group of diffeomorphisms”, Russian Math. Surveys, 30:6 (1975), 1–50  mathnet  crossref  mathscinet  zmath
    2. A. D. Gvishiani, “Canonical representations of the group $G=SL(2)$ over non-Archimedean fields and related representations of the group $G^X$”, Funct. Anal. Appl., 9:3 (1975), 203–214  mathnet  crossref  mathscinet  zmath
    3. K. R. Parthasarathy, K. Schmidt, “A new method for constructing factorisable representations for current groups and current algebras”, Comm Math Phys, 50:2 (1976), 167  crossref  mathscinet  zmath  adsnasa
    4. R F Streater, J Phys A Math Gen, 10:2 (1977), 261  crossref  mathscinet  zmath  adsnasa
    5. P Delorme, “Irreductibilité de certaines représentations de G(x)”, Journal of Functional Analysis, 30:1 (1978), 36  crossref
    6. I. B. Frenkel, V. G. Kac, “Basic representations of affine Lie algebras and dual resonance models”, Invent math, 62:1 (1980), 23  crossref  mathscinet  zmath  adsnasa  isi
    7. R. S. Ismagilov, “Representations of the group of smooth mappings of a segment into a compact Lie group”, Funct. Anal. Appl., 15:2 (1981), 134–135  mathnet  crossref  mathscinet  zmath  isi
    8. A. M. Vershik, S. V. Kerov, “Asymptotic theory of characters of the symmetric group”, Funct. Anal. Appl., 15:4 (1981), 246–255  mathnet  crossref  mathscinet  zmath  isi
    9. A. M. Vershik, S. I. Karpushev, “Cohomology of groups in unitary representations, the neighborhood of the identity, and conditionally positive definite functions”, Math. USSR-Sb., 47:2 (1984), 513–526  mathnet  crossref  mathscinet  zmath
    10. Yu. A. Neretin, “The complementary series of representations of the group of diffeomorphisms of the circle”, Russian Math. Surveys, 37:2 (1982), 229–230  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. Joachim Erven, Bernd-Jürgen Falkowski, “Low-order cohomology of semi-simple Lie groups with applications to continuous tensor products and current groups”, Acta Appl Math, 1:4 (1983), 333  crossref  mathscinet  zmath  isi
    12. A. M. Vershik, I. M. Gel'fand, M. I. Graev, “A commutative model of representation of the group of flows $SL(2,\mathbb{R})^X$ that is connected with a unipotent subgroup”, Funct. Anal. Appl., 17:2 (1983), 137–139  mathnet  crossref  mathscinet  zmath  isi
    13. S Albeverio, R Høegh-Krohn, D Testard, A Vershik, “Factorial representations of path groups”, Journal of Functional Analysis, 51:1 (1983), 115  crossref
    14. M. A. Lifshits, “Invariant measures generated by random fields with independent values”, Funct. Anal. Appl., 19:4 (1985), 329–330  mathnet  crossref  mathscinet  zmath  isi
    15. Yu. A. Neretin, “Representations of complementary series entering discretely in tensor products of unitary representations”, Funct. Anal. Appl., 20:1 (1986), 68–70  mathnet  crossref  mathscinet  zmath  isi
    16. A. Vourdas, “Analytic representations in the unit disk and applications to phase states and squeezing”, Phys Rev A, 45:3 (1992), 1943  crossref  adsnasa  isi
    17. Yu. A. Neretin, “Categories of bistochastic measures, and representations of some infinite-dimensional groups”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 197–219  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    18. Jean Marion, “Cylindrical representations of some infinite dimensional nuclear Lie groups”, Bull Austral Math Soc, 46:2 (1992), 295  crossref  mathscinet  zmath  isi
    19. I. M. Gel'fand, M. I. Graev, “Projective Representations of the Current Group $\operatorname{SU}(1,1)^X$”, Funct. Anal. Appl., 27:4 (1993), 275–277  mathnet  crossref  mathscinet  zmath  isi
    20. S. Albeverio, B. Torresani, “Some remarks on representations of jet groups and gauge groups”, J Math Phys (N Y ), 35:9 (1994), 4897  crossref  mathscinet  zmath  adsnasa  isi
    21. V. I. Arnol'd, M. Sh. Birman, I. M. Gel'fand, I. A. Ibragimov, S. V. Kerov, A. A. Kirillov, O. A. Ladyzhenskaya, G. A. Leonov, A. A. Lodkin, S. P. Novikov, Ya. G. Sinai, M. Z. Solomyak, L. D. Faddeev, “Anatolii Moiseevich Vershik (on his sixtieth birthday)”, Russian Math. Surveys, 49:3 (1994), 207–221  mathnet  crossref  mathscinet  adsnasa  isi
    22. A Vourdas, A Wünsche, J Phys A Math Gen, 31:46 (1998), 9341  crossref  mathscinet  zmath  adsnasa  isi
    23. Yehuda Shalom, “Bounded generation and Kazhdan’s property (T)”, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 90:1 (1999), 145  crossref  mathscinet  zmath
    24. G. Van Dijk, V.F. Molchanov, “Tensor products of maximal degenerate series representations of the group SL(n, R)”, Journal de Mathématiques Pures et Appliquées, 78:1 (1999), 99  crossref
    25. Yu. A. Neretin, “Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants”, Mosc. Math. J., 1:2 (2001), 157–220  mathnet  mathscinet  zmath  elib
    26. Natalia Tsilevich, Anatoly Vershik, Marc Yor, “An Infinite-Dimensional Analogue of the Lebesgue Measure and Distinguished Properties of the Gamma Process”, Journal of Functional Analysis, 185:1 (2001), 274  crossref
    27. Yu. A. Neretin, “The action of an overalgebra on the Plancherel decomposition and shift operators in the imaginary direction”, Izv. Math., 66:5 (2002), 1035–1046  mathnet  crossref  crossref  mathscinet  zmath  elib
    28. A. M. Vershik, N. V. Tsilevich, “Fock factorizations, and decompositions of the $L^2$ spaces over general Lévy processes”, Russian Math. Surveys, 58:3 (2003), 427–472  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    31. V. F. Molchanov, “Canonical Representations and Overgroups for Hyperboloids”, Funct. Anal. Appl., 39:4 (2005), 284–295  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    32. A Vourdas, “Analytic representations in quantum mechanics”, J Phys A Math Gen, 39:7 (2006), R65  crossref  mathscinet  zmath  adsnasa  isi
    33. A. M. Vershik, “On F. A. Berezin and his work on representations of current groups”, J. Math. Sci. (N. Y.), 141:4 (2007), 1385–1389  mathnet  crossref  mathscinet  zmath  elib
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    36. V. F. Molchanov, “Canonical representations on two-sheeted hyperboloids”, J. Math. Sci. (N. Y.), 141:4 (2007), 1432–1451  mathnet  crossref  mathscinet  zmath  elib  elib
    37. A. M. Vershik, “Does There Exist a Lebesgue Measure in the Infinite-Dimensional Space?”, Proc. Steklov Inst. Math., 259 (2007), 248–272  mathnet  crossref  mathscinet  zmath  elib  elib
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    40. Vershik, AM, “Invariant measures for the continual Cartan subgroup”, Journal of Functional Analysis, 255:9 (2008), 2661  crossref  mathscinet  zmath  isi  elib
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    42. Molchanov V.F., Artemov A.A., Grosheva L.I., “Kanonicheskie i granichnye predstavleniya”, Vestn. Tambovskogo un-ta. Ser.: Estestvennye i tekhnicheskie nauki, 14:6-3 (2009), 1367–1425  elib
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    44. 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
    45. Anton M. Zeitlin, “Unitary representations of a loop group, Wiener measure and Γ-function”, Journal of Functional Analysis, 2012  crossref
    46. I.B.. Frenkel, A.M.. Zeitlin, “On The Continuous Series for
      $${{\widehat{sl(2,{\mathbb{R}})}}}$$
      s l ( 2 , R ) ^”, Commun. Math. Phys, 2013  crossref
    47. 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
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    49. Lytvynov E., “Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations”, Commun. Math. Phys., 351:2 (2017), 653–687  crossref  mathscinet  zmath  isi  scopus
    50. 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
    51. Kosyak A., “Regular, Quasi-Regular and Induced Representations of Infinite-Dimensional Groups”, Regular, Quasi-Regular and Induced Representations of Infinite-Dimensional Groups, Ems Tracts in Mathematics, 29, European Mathematical Soc, 2018, 1–555  zmath  isi
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