The restriction of \(A_ q(\lambda)\) to reductive subgroups. II. (English) Zbl 0828.22030

[For Part I cf. ibid. 69, 262-267 (1993).]
In this paper we continue the investigation of the restriction of irreducible unitary representations of real reductive groups, with emphasis on the discrete decomposability. We recall that a representation \(\pi\) of a reductive Lie group \(G\) on a Hilbert space \(V\) is \(G\)- admissible if \((\pi, V)\) is decomposed into a discrete Hilbert direct sum with finite multiplicities of irreducible representations of \(G\). The same terminology is used for a \(({\mathfrak g}, K)\)-module on a pre-Hilbert space, if its completion is \(G\)-admissible.
Let \(H\) be a reductive subgroup of a real reductive Lie group \(G\), and \((\pi,V)\) an irreducible unitary representation of \(G\). The restriction \((\pi_{|H}, V)\) is decomposed uniquely into irreducible unitary representations of \(H\), which may involve a continuous spectrum if \(H\) is noncompact. In Part I of this paper and [Invent. Math. 117, 181-205 (1994)], we have posed the problem to single out the triplet \((G, H, \pi)\) such that the restriction of \((\pi_{|H}, V)\) is \(H\)- admissible, together with some application to harmonic analysis on homogeneous spaces. The purpose of this paper is to give new insight of such a triplet \((G, H, \pi)\) from view points of algebraic analysis. In particular, we will give a sufficient condition on the triplet \((G, H, \pi)\) for the \(H\)-admissible restriction as a generalization of [the author, loc. cit.] to arbitrary \(H\), and also present an obstruction for the \(H\)-admissible restriction.


22E99 Lie groups
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