§0. Introduction Interesting classes of (g, i^)-modules are often described naturally in terms of coho- mologically induced representations in various settings such as unitary highest weight modules, the theory of dual reductive pairs, discrete series for semisimple symmetric spaces, and etc. These have been stimulating the study of algebraic properties of derived functor modules. Now an almost satisfactory theory on derived functor modules includ- ing a functorial property about unitarizability, has been developed in the 'good' range of parameters. However, actual interesting families of unitary representations contain possibly singular parameters, so we are still faced with subtle problems such as finding conditions for non-vanishing, irreducibility, or pairwise inequivalence of these (g, K)- modules (see Problem(0.6)/). This paper treats such delicate algebraic properties of Zuckerman's derived functor modules with singular parameters which arise from discrete series for indefinite Stiefel manifolds U(p, q\ F)/U(p-m, q\ F) (p 2m, F = K, C or M). Some of them are isomorphic to "unipotent" representations in the philosophy of Arthur (e.g. [1], §5), and some are out of the range given by Vogan [29] for his unitarizability theorem. We should also remark that the unitary dual of a pseudo-orthogonal group U(p, q F) has not been classified yet, except in lower rank cases. Let G be a connected real reductive linear Lie group and if be a closed subgroup which is reductive in G. A homogeneous space G/H carries a G-invariant measure, so we have a natural unitary representation of G on the Hilbert space L2(G/H). Harish- Chandra modules of finite length realized in L2(G/H) are called discrete series for G/H. These play a fundamental role in harmonic analysis on G/H. If H is noncompact, Received by the editors 6 March, 1990 and, in revised form 30 April 1990. Partially supported by Grant-in-Aid for Scientific Research (No.01740081). 1
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