Canonical and boundary representations on the Lobachevsky plane (Q1862241)

The authors study canonical representations of the Lobachevsky plane. The representations are labeled by a complex parameter \(\lambda\). If \(\lambda\in (-3/2,0)\) then they coincide with the Vershik-Gelfand-Graev canonical unitary (with respect to the Berezin form) representations on a Hermitian symmetric space. But generally they are not unitary. The related boundary representations are introduced. The associated Poisson transform and the associated Fourier transform are described. A decomposition of the boundary and the canonical representation into irreducible components is proved. The decomposition is connected with the meromorphic structure of the Poisson and the Fourier transforms.