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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.

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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