an:06264539
Zbl 1317.22010
Kobayashi, Toshiyuki; Oshima, Toshio
Finite multiplicity theorems for induction and restriction
EN
Adv. Math. 248, 921-944 (2013).
0001-8708
2013
j
22E46 14M27
real reductive group; admissible representation; multiplicity; hyperfunction; unitary representation; spherical variety; symmetric space
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.