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)


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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