Finite multiplicity theorems

Details Finite multiplicity theorems Finite multiplicity theorems

2011-08-17

arxiv.org/abs/1108.3477v2

Mathematics

Representation Theory

26 pages

Scientific paper

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 $\operatorname{Ind}_H^G\tau$ from irreducible representations $\tau$ of a closed subgroup $H$. As corollaries, we establish geometric criteria for finiteness of the dimension of $\Hom_G(\pi,\Ind_H^G \tau)$ (induction) and of $\Hom_H(\pi|_H,\tau)$ (restriction) by means of the real flag variety $G/P$, and criteria for uniform boundedness of these multiplicities by means of the complex flag variety.

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