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Canonical and Boundary Representations on the Lobachevsky Plane

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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.

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  1. Tambov State University, Internatsionalnaya 33, Tambov, 392622, Russia

    V. F. Molchanov & L. I. Grosheva

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  1. V. F. Molchanov
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  2. L. I. Grosheva
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Molchanov, V.F., Grosheva, L.I. Canonical and Boundary Representations on the Lobachevsky Plane. Acta Applicandae Mathematicae 73, 59–77 (2002). https://doi.org/10.1023/A:1019770619285

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