Canonical Representations Related to Hyperbolic Spaces

We extend the notion of canonical representation, introduced by A. M. Vershik, I. M. Gelʹfand 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.