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Canonical Representations on Lobachevsky Spaces: An Interaction with an Overalgebra

Authors:	V F Molchanov

Institution:	(1) Tambov State University, Internatsionalnaya 33, Tambov, 392000, Russia

Abstract:	We define canonical representations R _λ, $\lambda\in\Bbb{C}$ , for the Lobachevsky space ℒ=G/K of dimension n−1 where G=SO₀(n−1,1), K=SO(n−1), as the restriction to G of maximal degenerate series representations of the overgroup $\widetilde{G}=\mathrm{SL}(n,\Bbb{R})$ . We determine explicitly the interaction of Lie operators of $\widetilde{G}$ with operators intertwining canonical representations and representations of G associated with a cone. Supported by the Russian Foundation for Basic Research: grants No. 05-01-00074a and No. 05-01-00001a, the Netherlands Organization for Scientific Research (NWO): grant 047-017-015, the Scientific Program “Devel. Sci. Potent. High. School”: project RNP.2.1.1.351 and Templan No. 1.2.02.

Keywords:	Lie groups Lie algebras Symmetric spaces Lobachevsky spaces Hyperboloids Canonical representations Berezin form Poisson and Fourier transforms Boundary representations
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