A generalized uniformly bounded multiplicity theorem (Developments in Representation Theory and Related Topics)

Abstract

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 T. Kobayashi and T. 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 He-orbits on Gc/B is finite, where Gc, He are complexifications of G, H, respectively, and B is a Borel subgroup of Ge. 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 Heorbits on Gc/Qc is finite. For the proof of this claim, we also prove the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic Dx-modules.