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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. 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.
Received: January 20, 2012; in revised form March 25, 2012
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
Linking options:
https://www.mathnet.ru/eng/mmj500 https://www.mathnet.ru/eng/mmj/v13/i2/p345
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