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Let P be a minimal parabolic subgroup of a real reductive Lie group G and H a closed subgroup of G. Then it is proved by Kobayashi and Oshima that the regular representation C ∞ (G / H) contains each irreducible representation of G at most finitely many times if the number of H -orbits on G / P is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of H ℂ -orbits on G ℂ / B is finite, where G ℂ , H ℂ are complexifications of G , H , respectively, and B is a Borel subgroup of G ℂ . In this paper, we prove that the multiplicities of the representations of G induced from a parabolic subgroup Q in the regular representation on G / H are uniformly bounded if the number of H ℂ -orbits on G ℂ / Q ℂ is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic X -modules. [ABSTRACT FROM AUTHOR]

Abstract: We find upper and lower bounds of the multiplicities of irreducible admissible representations π of a semisimple Lie group G occurring in the induced representations from irreducible representations τ of a closed subgroup H. As corollaries, we establish geometric criteria for finiteness of the dimension of (induction) and of (restriction) by means of the real flag variety , and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety. [Copyright &y& Elsevier]