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

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.