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Indefinite intertwining operators. II. (English) Zbl 0651.22008

Harmonic analysis, symmetric spaces and probability theory, Cortona/Italy 1984, Symp. Math. 29, 1-46 (1987).
[For the entire collection see Zbl 0633.00012. For Part I cf. Proc. Natl. Acad. Sci. USA 81, 1272-1275 (1984; Zbl 0539.22009).]
The major part of the paper is devoted to a complete classification of the unitary irreducible representations of SU(p,2). The result it to complicated to be stated here, but the principle underlying the classification may be described as follows. Let G be a connected semisimple linear Lie group, J(S,\(\sigma\),\(\nu)\) the Langlands quotient of the induced representation U(S,\(\sigma\),\(\nu)\). Assume G has a compact Cartan subgroup B and that all noncompact roots are short. Then there is \(\omega_ 0\in K\) normalizing A so that \(Ad(\omega_ 0)\) acts by -1 on a. For real \(\nu\), \(\omega_ 0\nu ={\bar \nu}\) which insures that J(S,\(\sigma\),\(\nu)\) admits an invariant Hermitian form, necessarily given by the intertwining operator \(\sigma (\omega_ 0)A_ S(w_ 0,\sigma,\nu)\). Let \(\tau_{\Lambda}\) be a minimal K-type of U(S,\(\sigma\),\(\nu)\). Since \(\tau_{\Lambda}\) occurs with multiplicity one, the operator may be normalized so as to be \(=1\) on this K-type for all \(\nu\). Let T(\(\nu)\) be this normalized operator. If there is another K-type on which T(\(\nu)\) is not positive semidefinite, then the Hermitian form on J(S,\(\sigma\),\(\nu)\) cannot be semidefinite and J(S,\(\sigma\),\(\nu)\) cannot be unitary. The authors state four general theorems which allow to decide this question in certain cases. (The proof of these theorems is deferred to another paper.) They apply these results to the classification of the irreducible unitary representations of SU(p,2).
Reviewer: W.Rossmann

MSC:

22E46 Semisimple Lie groups and their representations
22D10 Unitary representations of locally compact groups