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Singular unitary representations and discrete series for indefinite Stiefel manifolds $$U(p,q;\mathbb{F})/U(p-m,q;\mathbb{F})$$. (English) Zbl 0752.22007
Mem. Am. Math. Soc. 462, 106 p. (1992).
Author’s abstract: This paper treats the relatively singular part of the unitary dual of pseudo-orthogonal groups over $$\mathbb{F}=\mathbb{R}$$, $$\mathbb{C}$$ and $$\mathbb{H}$$. These representations arise from discrete series for indefinite Stiefel manifolds $$U(p,q;\mathbb{F})/U(p-m,q;\mathbb{F})$$, $$2m\leq p$$. Thanks to the duality theorem between $${\mathcal D}$$-module construction and Zuckerman’s derived functor modules (ZDF-modules), these discrete series are naturally described in terms of ZDF-modules with possibly singular parameters. Some techniques including a new $$K$$-type formula are offered to find the explicit condition deciding whether the corresponding ZDF- modules vanish or not. We also investigate the irreducibility and pairwise inequivalence among these ZDF-modules. Although our concern is limited to the discrete series, our approach is purely algebraic and applicable to a less special setting. It is an interesting phenomenon that our discrete series sometimes give a sharper condition for unitarizability of ZDF-modules than those given by Vogan (1984) algebraically. This phenomenon does not occur in the case of discrete series for group manifolds or semisimple symmetric spaces.
Reviewer: F.Rouvière (Nice)

##### MSC:
 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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