## Integral models of unitary representations of current groups with values in semidirect products

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Vershik, A. M.

Graev, M. I.

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##### Abstract

##### Description

We describe a general construction of irreducible unitary representations of
the group of currents with values in the semidirect product of a locally
compact subgroup $P_0$ and a one-parameter group ${\mathbb R {}}^*_+=\{r:r>0\}$
of automorphisms of $P_0$. This construction is determined by a a faithful
unitary representation of $P_0$ (canonical representation) whose images under
the action of the group of automorphisms tend to the identity representation as
$r\to 0$. We apply this construction to the groups of currents of the maximal
parabolic subgroups of the groups of motions of the $n$-dimensional real and
complex Lobachevsky spaces. The obtained representations of the groups of
parabolic currents can be uniquely extended to the groups of currents with
values in the semisimple groups O(n,1) and U(n,1). This gives a new description
of the representations of the groups of currents of these groups constructed in
the 70s and realized in the Fock space. The key role in our construction is
played by the so-called special representation of the parabolic subgroup $P$
and the remarkable $\sigma$-finite measure (Lebesgue measure) $\mathcal L$ in
the space of distributions.

Comment: 13 pp, Ref.18

Comment: 13 pp, Ref.18

##### Keywords

Mathematics - Representation Theory, Mathematics - Functional Analysis, 22D10,43A65