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  "body": "\n## Summary\n\nAn admissible representation of a reductive group is one in which each irreducible representation of a maximal compact subgroup occurs only finitely many times.\nThe condition singles out the representations that can be studied by algebraic means, through their [[Harish-Chandra modules|2MKHH636C3]].\nEvery irreducible unitary representation of a real reductive group is admissible, a theorem of Harish-Chandra.\nOver $p$-adic fields the analogous notion asks that the vectors fixed by each compact open subgroup form a finite-dimensional space.\n\n## Explanations\n\n### L1 human\n\nRepresentations of non-compact groups such as $\\mathrm{SL}_2(\\R)$ usually act on infinite-dimensional spaces.\nRestrict the action to the compact subgroup of rotations $\\mathrm{SO}(2)$, and the space breaks into simple pieces, one for each \"frequency\" $e^{in\\theta}$.\nAdmissible means that each frequency appears only a finite number of times.\nSo the representation is infinite, but it is infinite in a controlled way.\n",
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