Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations.

*(English)*Zbl 1439.22026Summary: Let \(G\) be a real reductive Lie group and \(H\) a closed subgroup. T. Kobayashi and T. Oshima established a finiteness criterion of multiplicities of irreducible \(G\)-modules occurring in the regular representation \(C^{\infty}(G/H)\) by a geometric condition, referred to as real sphericity, namely, \(H\) has an open orbit on the real flag variety \(G/P\). This note discusses a refinement of their theorem by replacing a minimal parabolic subgroup \(P\) with a general parabolic subgroup \(Q\) of \(G\), where a careful analysis is required because the finiteness of the number of \(H\)-orbits on the partial flag variety \(G/Q\) is not equivalent to the existence of \(H\)-open orbit on \(G/Q\).

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

20G05 | Representation theory for linear algebraic groups |

53C30 | Differential geometry of homogeneous manifolds |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14M27 | Compactifications; symmetric and spherical varieties |

##### Keywords:

degenerate principal series; multiplicity; spherical variety; intertwining operators; real spherical; reductive group
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\textit{T. Tauchi}, Proc. Japan Acad., Ser. A 95, No. 7, 75--79 (2019; Zbl 1439.22026)

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