
This article is cited in 1 scientific paper (total in 2 paper)
Special representations of nilpotent Lie groups and the associated Poisson representations of current groups
A. M. Vershik^{a}, M. I. Graev^{b} ^{a} St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg 191023, Russia
^{b} Institute for System Studies, 361 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 wellknown Fock space. The construction used the socalled singular
representation of the coefficient group, in which the first cohomology of this group is nontrivial.
In this paper we give a new construction using a special property of onedimensional extension of nilpotent groups, which allows immediately to describe the singular representation, and then to apply the quasiPoisson 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/10.17323/160945142013132345360
Full text:
http://www.mathjournals.org/.../2013013002007.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
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\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 345360
\mathnet{http://mi.mathnet.ru/mmj500}
\crossref{https://doi.org/10.17323/160945142013132345360}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3134910}
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http://mi.mathnet.ru/eng/mmj500 http://mi.mathnet.ru/eng/mmj/v13/i2/p345
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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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