Admissible representation

Summary

An admissible representation of a reductive group is one in which each irreducible representation of a maximal compact subgroup occurs only finitely many times. The condition singles out the representations that can be studied by algebraic means, through their Harish-Chandra modules. Every irreducible unitary representation of a real reductive group is admissible, a theorem of Harish-Chandra. Over $p$-adic fields the analogous notion asks that the vectors fixed by each compact open subgroup form a finite-dimensional space.