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Discrete decomposability of restriction of \(A_{\mathfrak q}(\lambda)\) with respect to reductive subgroups and its applications. (English) Zbl 0826.22015
Sei \(G\) eine lokalkompakte Gruppe vom Typ 1 und \((\pi, V)\) eine unitäre Darstellung von \(G\) auf dem Hilbertraum \(V\). \((\pi,V)\) heißt diskret zerlegbar, falls sie unitär äquivalent zu \(\sum^\oplus_m (\tau) (\tau, H_\tau)\), \((\tau, H_\tau)\) irreduz., ist (mit \(m(\tau) := \dim_\mathbb{C} \text{Hom}_G (H_\tau, V)\)). Man nennt \((\pi, V)\) \(G\)- zulässig, falls die Multiplizität \(m(\tau) < \infty\). Ein erstes Ergebnis der vorliegenden Arbeit ist, daß die \(K\)-Zulässigkeit von \((\pi|_K, V)\) für eine Untergruppe \(K\) die \(G\)-Zulässigkeit von \((\pi, V)\) impliziert. Hauptresultat ist eine hinreichende Bedingung für die \(G'\)-Zulässigkeit der Zuckermanschen induzierten Darstellung \(\overline {A_q (\lambda)}|_{G'}\) für den Fall symmetrischer Paare \((G,G')\) und holomorpher Einbettungen. Als Anwendung ergeben sich neue Erkenntnisse über diskrete Serien für gewisse pseudoriemannsche (nicht-symmetrische) sphärische homogene Räume.

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22D10 Unitary representations of locally compact groups
22D30 Induced representations for locally compact groups
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