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Finite multiplicity theorems for induction and restriction. (English) Zbl 1317.22010
Summary: 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 \(\mathrm{Ind}_H^G{\tau}\) from irreducible representations \({\tau}\) of a closed subgroup \( H\). As corollaries, we establish geometric criteria for finiteness of the dimension of \(\mathrm{Hom}_G({\pi}, \mathrm{Ind}_H^G{\tau})\) (induction) and of \(\mathrm{Hom}_H({\pi}|_H,{\tau})\) (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.

MSC:
22E46 Semisimple Lie groups and their representations
14M27 Compactifications; symmetric and spherical varieties
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