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Unitary derived functor modules with small spectrum. (English) Zbl 0568.22007
This article has to do with the problem of finding the unitary dual of a real semisimple Lie group G. In order to describe all irreducible unitary representations of G, one needs Zuckerman’s derived functor parabolic induction. The difficulty is that this construction yields irreducible representations which may or may not be unitary. The main general results of this paper give sufficient conditions so that derived functor parabolic induction yields a unitary representation. These results were announced earlier by the same authors [Proc. Natl. Acad. Sci. USA 80, 7047-7050 (1983; Zbl 0527.22007)].
More than a third of the article consists of applications of the general theory to specific settings. These applications include: (1) The authors construct a unitary representation of SO(p,q) with \(p+q\) even. It is multiplicity free as a \(k\)-module and the highest weights of the \(k\)- submodules all lie along a single line. For this reason it is called a ladder representation. Here \(k\) is the complexification of the Lie algebra of the subgroup K which is the maximal connected subgroup of G whose image in G/center is compact. (2) They construct a family of ladder representations for each of the groups Sp(r,s). (3) Several families of unitary reprsentations of Sp(n,\({\mathbb{R}})\) are constructed. (4) The main results are used to give an alternate proof of unitarity for the Speh representations of SL(2n,\({\mathbb{R}})\). Analogous representations of \(SU^*(2n)\) are also described. (5) Unitarity results are obtained for the analytic continuation of certain discrete series representations of G.
Reviewer: Th.Farmer

22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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