Finite multiplicity theorems for induction and restriction

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