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Russian Mathematical Surveys, 2006, Volume 61, Issue 5, Pages 799–884
DOI: https://doi.org/10.1070/RM2006v061n05ABEH004356
(Mi rm3389)
 

This article is cited in 14 scientific papers (total in 16 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
References:
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.
Received: 10.05.2006
Russian version:
Uspekhi Matematicheskikh Nauk, 2006, Volume 61, Issue 5(371), Pages 3–88
DOI: https://doi.org/10.4213/rm3389
Bibliographic databases:
Document Type: Article
UDC: 517.5
MSC: Primary 22E65, 22D10; Secondary 20G20
Language: English
Original paper language: Russian
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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\paper Structure of the complementary series and special representations of the groups $O(n,1)$ and~$U(n,1)$
\jour Uspekhi Mat. Nauk
\yr 2006
\vol 61
\issue 5(371)
\pages 3--88
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\jour Russian Math. Surveys
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\vol 61
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\pages 799--884
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  • This publication is cited in the following 16 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    References:101
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