Canonical and Boundary Representations on the Lobachevsky Plane

Research paper by V. F. Molchanov, L. I. Grosheva

Indexed on: 01 Aug '02Published on: 01 Aug '02Published in: Acta applicandae mathematicae

Abstract

Canonical representations on Hermitian symmetric spaces G/K were introduced by Vershik, Gelfand and Graev and Berezin. They are unitary. We study canonical representations in a wider sense. In this paper we restrict ourselves to a crucial example – the Lobachevsky plane: G=SU(1,1), K=U(1). Canonical representations are labelled by the complex parameter λ (Vershik–Gelfand–Graev's representations correspond to −3/2<λ<0). We decompose the canonical representations into irreducible components. The decomposition includes boundary representations generated by the canonical representations. So we study these boundary representations themselves. The decomposition of boundary representations is closely connected with the meromorphic structure of Poisson and Fourier transforms associated with canonical representations. In particular, second-order poles give second-order Jordan blocks. Finally, we give a full decomposition of the Berezin transform using ‘generalized powers’ (Pochhammer symbols) instead of usual powers of λ.