This work studies the canonical representations (Berezin representations) for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces G/H with G = SL(n,R),H = GL(n-1,R) . For Hermitian symmetric spaces G/K, canonical representations were introduced by Berezin and Vershik-Gelfand-Graev. They are unitary with respect to some invariant non-local inner product (the Berezin form). We consider canonical representations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distributions. For our spaces G/H, the canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. We decompose the canonical representations into irreducible constituents and decompose boundary representations.
G. van Dijk and V. F. Molchanov, “Tensor Products of Maximal Degenerate Series Representations of the Group ,” Journal de Mathématiques Pures et Appliquées, Vol. 78, No. 1, 1999, pp. 99-119. http://dx.doi.org/10.1016/S0021-7824(99)80011-7
G. van Dijk and V. F. Molchanov, “The Berezin form for Rank One Para-Hermitian Symmetric Spaces,” Journal de Mathématiques Pures et Appliquées, Vol. 77, No. 8, 1998, pp. 747-799. http://dx.doi.org/10.1016/S0021-7824(98)80008-1