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Uspekhi Mat. Nauk, 2006, Volume 61, Issue 5(371), Pages 3–88 (Mi umn3389)  

This article is cited in 12 scientific papers (total in 14 papers)

Structure of the complementary series and special representations of the groups $O(n,1)$ and $U(n,1)$

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: This is a survey of several models (including new models) of irreducible complementary series representations and their limits, special representations, for the groups $SU(n,1)$ and $SO(n,1)$. 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 (that is, 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 of the group with coefficients in these representations is non-trivial. For technical reasons it is more convenient to consider the groups $O(n,1)$ and $U(n,1)$, and most of this paper is devoted to the group $U(n,1)$.
The main emphasis is on the so-called commutative models of special and complementary series representations: in these models, the maximal unipotent subgroup is represented by multipliers in the case of $O(n,1)$, and by the canonical model of the Heisenberg representations in the case of $U(n,1)$. Earlier, these models were studied only for the group $ SL(2,\mathbb R)$. They are especially important for the realization of non-local representations of current groups, which will be considered elsewhere.
Substantial use is made of the ‘denseness’ of the irreducible representations under study for the group $SO(n,1)$: their restrictions to the maximal parabolic subgroup $P$ are equivalent irreducible representations. Conversely, in order to extend an irreducible representation of $P$ to a representation of $SO(n,1)$, it is necessary to determine only one involution. For the group $U(n,1)$, the situation is similar but slightly more complicated.

DOI: https://doi.org/10.4213/rm3389

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English version:
Russian Mathematical Surveys, 2006, 61:5, 799–884

Bibliographic databases:

UDC: 517.5
MSC: Primary 22E65, 22D10; Secondary 20G20
Received: 10.05.2006

Citation: A. M. Vershik, M. I. Graev, “Structure of the complementary series and special representations of the groups $O(n,1)$ and $U(n,1)$”, Uspekhi Mat. Nauk, 61:5(371) (2006), 3–88; Russian Math. Surveys, 61:5 (2006), 799–884

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. 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
    2. A. M. Vershik, M. I. Graev, “Integral Models of Representations of Current Groups”, Funct. Anal. Appl., 42:1 (2008), 19–27  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. M. Vershik, M. I. Graev, “Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products”, Funct. Anal. Appl., 42:4 (2008), 279–289  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. A. M. Vershik, I. M. Gel'fand, S. G. Gindikin, A. A. Kirillov, G. L. Litvinov, V. F. Molchanov, Yu. A. Neretin, V. S. Retakh, “Mark Iosifovich Graev (to his 85th brithday)”, Russian Math. Surveys, 63:1 (2008), 173–188  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. 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
    6. Speh B., Venkataramana T.N., “Discrete components of some complementary series”, Forum Math., 23:6 (2011), 1159–1187  crossref  mathscinet  zmath  isi  elib  scopus
    7. 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
    8. A. M. Vershik, M. I. Graev, “Osobye predstavleniya grupp $U(\infty,1)$ i $O(\infty,1)$ i svyazannye s nimi predstavleniya grupp tokov $U(\infty,1)^X$ i $O(\infty,1)^X$ v kvazipuassonovom prostranstve”, Funkts. analiz i ego pril., 46:1 (2012), 1–12  mathnet  crossref  mathscinet  zmath  elib
    9. A. M. Vershik, M. I. Graev, “Special representations of nilpotent Lie groups and the associated Poisson representations of current groups”, Mosc. Math. J., 13:2 (2013), 345–360  mathnet  crossref  mathscinet
    10. 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
    11. 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
    12. Hilgert J., Kobayashi T., Mollers J., “Minimal Representations Via Bessel Operators”, J. Math. Soc. Jpn., 66:2 (2014), 349–414  crossref  mathscinet  zmath  isi  scopus
    13. Kobayashi T., Speh B., “Symmetry breaking for representations of rank one orthogonal groups”, Mem. Am. Math. Soc., 238:1126 (2015), 1+  crossref  mathscinet  isi
    14. 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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