We extend the notion of canonical representation, introduced by A. M. Vershik, I. M. Gel'fand and M. I. Graev (1982, in ''Representation Theory,'' Cambridge Univ. Press, Cambridge, U.K.), to all (classical) rank one semisimple Lie groups. We determine the spectral decomposition of the canonical representations in detail. (C) 1997 Academic Press. [References: 17]
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