Finite multiplicity theorems for induction and restriction

Abstract

We find upper and lower bounds of the multiplicities of irreducible admissible representations $\pi$ of a semisimple Lie group $G$ occurring in the induced representations $In{d}_H^G\tau$ from irreducible representations $\tau$ of a closed subgroup $H$. As corollaries, we establish geometric criteria for finiteness of the dimension of $Ho{m}_G(\pi ,In{d}_H^G\tau )$ (induction) and of $Ho{m}_H(\pi {|}_H,\tau )$ (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.

to appear in Advances in Mathematics

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