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.

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:

22E99 | Lie groups |

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\textit{T. Kobayashi}, Proc. Japan Acad., Ser. A 71, No. 1, 24--26 (1995; Zbl 0828.22030)

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##### References:

[1] | W. Borho and J. L. Brylinski: Differential operators on homogeneous spaces. I. Invent. Math., 69, 437-476 (1982). · Zbl 0504.22015 |

[2] | M. Kashiwara, T. Kawai and T. Kimura: Foundations of Algebraic Analysis. Princeton Math. Series, 37(1986). · Zbl 0605.35001 |

[3] | M. Kashiwara and M. Vergne: if-types and singular spectrum. Lect. Notes in Math., vol. 728, pp. 177-200 (1979). · Zbl 0411.22015 |

[4] | T. Kobayashi: Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds U(p, q ;F)/U(p - rn, q ;F). vol. 462, Memoirs A. M. S. (1992). · Zbl 0752.22007 |

[5] | T. Kobayashi: The restriction of Aq(A) to reductive subgroups. Proc. Japan Acad., 69A, 262-267 (1993). · Zbl 0826.22014 |

[6] | T. Kobayashi: Discrete decomposability of the restriction of Aq(X) with respect to reductive subgroups and its applications. Invent. Math., 117, 181-205 (1994). · Zbl 0826.22015 |

[7] | D. Vogan: Unitary Representations of Reductive Lie Groups. Ann. Math. Stud., 118, Princeton U. P. (1987). · Zbl 0626.22011 |

[8] | D. Vogan: Irreducibility of discrete series representations for semisimple symmetric spaces. Advanced Studies in Pure Math., 14, 191-221 (1988). · Zbl 0733.22008 |

[9] | D. Vogan: Associated varieties and unipotent representations. Harmonic Analysis on Reductive Lie Groups, vol. 101, Birkhauser pp. 315-388 (1991). · Zbl 0832.22019 |

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