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

TL;DR

This paper constructs irreducible unitary representations of current groups valued in semidirect products, extending known representations of parabolic subgroups of motion groups in Lobachevsky spaces.

Contribution

It introduces a new construction method for irreducible representations of current groups based on faithful unitary representations and automorphism limits, applied to parabolic subgroups of motion groups.

Findings

Constructed new irreducible unitary representations of current groups.

Extended representations to groups with values in semisimple groups O(n,1) and U(n,1).

Provided a new description of existing Fock space representations.

Abstract

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.