---
tag: SZV8Y806Y7
kind: concept
name: Admissible representation
msc: ["22E46", "22E50", "22E45"]
specializes: [7JA7WH92K0]
sources:
  - url: https://en.wikipedia.org/wiki/Admissible_representation
same_as:
  wikidata: Q4683806
  wikipedia: 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|2MKHH636C3]].
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.

## Explanations

### L1 human

Representations of non-compact groups such as $\mathrm{SL}_2(\R)$ usually act on infinite-dimensional spaces.
Restrict the action to the compact subgroup of rotations $\mathrm{SO}(2)$, and the space breaks into simple pieces, one for each "frequency" $e^{in\theta}$.
Admissible means that each frequency appears only a finite number of times.
So the representation is infinite, but it is infinite in a controlled way.