## Representations of pseudo-unitary groups associated with a cone.(English)Zbl 0941.22015

The representations of the pseudo-unitary group $$SU(p,q)$$, $$p,q\geq 2$$, associated with an isotropic cone are studied. These representations are important in constructing harmonic analysis on hyperbolic spaces. The main instrument used in this paper is the restriction to the maximal compact subgroup $$K=S(U(p) \times U(q))$$. The zonal spherical functions for $$K$$-types are obtained and the structure of representations associated with a cone (irreducibility, composition series, etc.) is studied. The intertwining operators and invariant Hermitian forms are determined in order to establish which representations are unitarizable. The author concludes that the representations in the first and second discrete series, the exceptional representation and the unit representation are unitarizable.
Reviewer: G.Zet (Iaşi)

### MSC:

 2.2e+47 Semisimple Lie groups and their representations
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