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Continuation of unitary derived functor modules out of the canonical chamber. (English) Zbl 0582.22013
The authors present a method for proving unitarizability for representations of a real semi-simple group G obtained from certain \({\mathfrak g}\)-modules of its Lie algebra \({\mathfrak g}\). The \({\mathfrak g}\)- modules considered are the Zuckerman derived functor modules obtained from generalized Verma modules and the method is that of an analytic continuation by means of varying the character of the Cartan subalgebra. This method relies on an earlier joint work of the authors with R. Parthasarathy and N. R. Wallach [Proc. Natl. Acad. Sci. USA 80, 7047-7050 (1983; Zbl 0527.22007)] where a general criterion for obtaining unitary representations that way was proved.
Here an account of a computational procedure is presented enabling an application of this general result to classical simple groups and, by means of appended computer programs, to exceptional groups \(E_ 6\), \(E_ 7\), \(E_ 8\). The new unitary representations obtained are tabulated in the appendix. Use is made of Jantzen’s irreducibility criterion [J. C. Jantzen, Math. Ann. 226, 53-65 (1977; Zbl 0372.17003)].
Reviewer: A.Strasburger

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E46 Semisimple Lie groups and their representations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22-04 Software, source code, etc. for problems pertaining to topological groups
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References:
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