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

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

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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• https://doi.org/10.4213/rm3389
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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
2. A. M. Vershik, M. I. Graev, “Integral Models of Representations of Current Groups”, Funct. Anal. Appl., 42:1 (2008), 19–27
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
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
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
6. Speh B., Venkataramana T.N., “Discrete components of some complementary series”, Forum Math., 23:6 (2011), 1159–1187
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
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
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
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
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
12. Hilgert J., Kobayashi T., Mollers J., “Minimal Representations Via Bessel Operators”, J. Math. Soc. Jpn., 66:2 (2014), 349–414
13. Kobayashi T., Speh B., “Symmetry breaking for representations of rank one orthogonal groups”, Mem. Am. Math. Soc., 238:1126 (2015), 1+
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
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