Canonical Representations and Overgroups for Hyperboloids of One Sheet and Lobachevsky Spaces
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Canonical Representations and Overgroups for Hyperboloids of One Sheet and Lobachevsky Spaces V. F. MOLCHANOV Tambov State University, Internatsionalnaya 33, 392622 Tambov, Russia. e-mail: [email protected] Abstract. We define canonical representations for the hyperboloid of one sheet X = G/H and for the Lobachevsky space L = G/K where G = SO0 (1, n − 1), H = SO0 (1, n − 2) and K = = SO(n − 1), as the restriction to G of representations associated with a cone of overgroups G = SO0 (1, n) for X and L respectively. We determine explicitly the interaction SO0 (2, n − 1) and G with operators intertwining canonical representations and representations of G of Lie operators of G associated with a cone. Mathematics Subject Classifications (2000): 43A85, 22E46, 53D55. Key words: Lie groups, Lie algebras, symmetric spaces, hyperboloids, canonical representations, Poisson and Fourier transforms, boundary representations.
Canonical representations on Hermitian symmetric spaces G/K were introduced by Berezin [1, 2] and Vershik, Gelfand and Graev [8] – for the purposes of quantization and quantum field theory. They are unitary with respect to some invariant non-local inner product (the Berezin form). We think that it is natural to consider canonical representations in a wider sense: to give up the condition of unitarity and let these representations act on more extensive spaces. The notion of canonical representation (in this wide sense) can be extended to other classes of semisimple symmetric spaces G/H , in particular, to para-Hermitian symmetric spaces, see [5]. Moreover, sometimes it is natural to consider at once several spaces G/Hi , possibly with different Hi , embedded as open G-orbits into a compact manifold , where G acts, so that is the closure of these orbits. In all these cases the canonical representation, denote it R, can be obtained as (irreducible, as a rule) of a the restriction to the group G of a representation R bigger group (“overgroup”) G. In this paper we consider the pseudo-orthogonal group G = SO0 (1, n − 1) and its hyperbolic homogeneous spaces in Rn : the hyperboloid of one sheet X = G/H Supported by grants of the Netherlands Organization for Scientific Research (NWO): 047-008-
009, the Russian Foundation for Basic Research (RFBR): 01-01-00100-a, the Minobr RF: E00-1.0156, the NTP “Univ. Rossii”: ur04.01.037.
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and the Lobachevsky space L = G/K, here H = SO0 (1, n − 2) and K = = SO0 (1, n) for X and = SO0 (2, n−1) and G SO(n−1). As overgroups we take G L respectively. We define canonical representations Rλ,ν for X and Rλ for L (here λ,ν or R λ of the correλ ∈ C, ν = 0, 1) as the restrictions to G of representations R associated with a cone. They act on functions on a compact sponding overgroups G manifold which is the direct product of two spheres of dimensions 1 and n−2 for X and the sphere of dimension n−1 for L. This manifold contains two copies of X or L as open G-orbits and is the closure of them. The canonical representat
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