## 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.

### MSC:

 2.2e+100 Lie groups
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### References:

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