T. Kobayashi and T. Oshima. Finite multiplicity theorems for induction and restriction. Advances in Mathematics 248 (2013), 921-944. DOI: 10.1016/j.aim.2013.07.015. arXiv:1108.3477..

We find upper and lower bounds of the multiplicities of irreducible admissible representations π of semisimple Lie groups G occurring in the induced representations IndHGτ from irreducible representations τ of closed subgroups H. As corollaries, we establish geometric criteria for finiteness of the dimension of HomG(π,IndHG τ) (induction) and of HomH(π|H,τ) (restriction) by means of the real flag variety G/P, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.

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