Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products

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 $P_0$ by a one-parameter group $\mathbb{R}^*_+=\{r:r>0\}$ of automorphisms of $P_0$. This construction is determined by 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 current groups of maximal parabolic subgroups in the groups of motions of the $n$-dimensional real and complex Lobachevsky spaces. The obtained representations of the current groups of parabolic subgroups uniquely extend to the groups of currents with values in the groups $O(n,1)$ and $U(n,1)$. This gives a new description of the representations, constructed in the 1970s and realized in the Fock space, of the current groups of the latter groups. The key role in our construction is played by the so-called special representation of the parabolic subgroup $P$ and a remarkable $\sigma$-finite measure (Lebesgue measure) $\mathcal L$ on the space of distributions.