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 Mosc. Math. J., 2013, Volume 13, Number 2, Pages 345–360 (Mi mmj500)

Special representations of nilpotent Lie groups and the associated Poisson representations of current groups

A. M. Vershika, M. I. Graevb

a St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg 191023, Russia
b Institute for System Studies, 36-1 Nakhimovsky pr., 117218 Moscow, Russia

Abstract: We describe models of representations of current groups for such semisimple Lie groups of rank 1 as $\mathrm O(n,1)$ and $\mathrm U(n,1)$, $n\ge1$.
This problem was posed in the beginning of the 70ies (Araki, Vershik–Gelfavd–Graev) and solved first for $\mathrm{SL}(2,\mathbb R)$, and then for all the above mentioned groups in the works of the three authors; the representations were realized in the well-known Fock space. The construction used the so-called singular representation of the coefficient group, in which the first cohomology of this group is non-trivial.
In this paper we give a new construction using a special property of one-dimensional extension of nilpotent groups, which allows immediately to describe the singular representation, and then to apply the quasi-Poisson model, which was constructed in previous works by the authors. First one constructs a representation of the current group of the $1$-dimensional extension of the nilpotent group; it is possible to show that this representation can be exteneded to the parabolic subgroup first, and then to the whole semisimple group.
As a result, one obtains a simple and clear proof of the irreducibility of the classical representation of current groups for semisimple groups.

Key words and phrases: current group, canonical representation, special representation.

DOI: https://doi.org/10.17323/1609-4514-2013-13-2-345-360

Full text: http://www.mathjournals.org/.../2013-013-002-007.html
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Bibliographic databases:

MSC: 22E27, 22E65, 46F25
Received: January 20, 2012; in revised form March 25, 2012
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Citation: 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

Citation in format AMSBIB
\Bibitem{VerGra13} \by A.~M.~Vershik, M.~I.~Graev \paper Special representations of nilpotent Lie groups and the associated Poisson representations of current groups \jour Mosc. Math.~J. \yr 2013 \vol 13 \issue 2 \pages 345--360 \mathnet{http://mi.mathnet.ru/mmj500} \crossref{https://doi.org/10.17323/1609-4514-2013-13-2-345-360} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3134910} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000317381000006} 

• http://mi.mathnet.ru/eng/mmj500
• http://mi.mathnet.ru/eng/mmj/v13/i2/p345

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This publication is cited in the following articles:
1. 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
2. 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