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Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations. (English) Zbl 1439.22026
Summary: 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
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References:
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