# Moscow Mathematical Journal

Volume 13, Issue 2, April–June 2013 pp. 345–360.

Special Representations of Nilpotent Lie Groups and the Associated Poisson Representations of Current Groups**Authors**: Anatoly M. Vershik (1) and Mark I. Graev (2)

**Author institution:**(1) St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg 191023, Russia

(2) Institute for System Studies, 36-1 Nakhimovsky pr., 117218 Moscow, Russia

**Summary:**

We describe models of representations of current groups for such semisimple
Lie groups of rank 1 as O(*n*,1) and U(*n*,1), *n*≥1.

This problem was posed in the beginning of the 70ies (Araki, Vershik–Gelfavd–Graev) and solved first for SL(2,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.

2010 Mathematics Subject Classification. 22E27, 22E65, 46F25.

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