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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
\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
\jour Russian Math. Surveys
\yr 1973
\vol 28
\issue 5
\pages 87--132

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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
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    13. S Albeverio, R Høegh-Krohn, D Testard, A Vershik, “Factorial representations of path groups”, Journal of Functional Analysis, 51:1 (1983), 115  crossref
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    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
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    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
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