Canonical representations associated to hyperbolic spaces II

AbstractIn this paper we determine explicit models for the unitary representations which occur discretely in the decomposition of canonical representations for hyperbolic spaces. This generalizes work of Gelfand, Graev and Vershik for the case SU(1,1).In a previous paper [1] we have introduced canonical representations πλ for hyperbolic spaces of Riemannian type, generalizing Berezin's definition for the case of Hermitian symmetric spaces. These unitary representations have a rich internal structure and prove that not only quasi-regular representations are important in harmonic analysis. They have an interesting decomposition into irreducible constituents: for large λ only unitary principal series representations occur, for small λ however a finite number of complementary series representations has to be added. A detailed description of the latter, as subrepresentations of πλ, was started in [1]. The present paper contains much more precise results. The classical notions of Fourier and Poisson transform play a dominant role and are treated in Section 4. In Section 5 the actual projection of a given function onto the complementary series is then given. Since the proof makes use of some results on maximal degenerate series representations of SL(n + 1), these representations are studied in Section 1 in some detail. As a by-product we obtain an alternative introduction of the canonical representations, see Sections 2 and 5.This paper has a considerable, but unevitable optical overlap with our joint paper [2] with V.F. Molchanov. The method described here has been introduced in a different setting by Molchanov, but the observation that it works also for hyperbolic spaces is new and surprising. To leave out parts of the proof would make this paper almost unreadable. We thank S.C. Hille and V.F. Molchanov for several discussions on these matters and for their encouragement to publish the results obtained in this paper