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  "body": "\n## Definition\n\nLet $G$ be a [[real reductive Lie group|42ZCE14AD2]] and let $K$ be a [[maximal compact subgroup|T2EB8R9HEV]] of $G$.\nLet $\\pi$ be a [[representation|WYGRXCG2RN]] of $G$ on a [[Hilbert space|HM2HHE3J51]] $V$ such that the map $G \\times V \\to V$, $(g, v) \\mapsto \\pi(g) v$, is continuous.\nThe representation $\\pi$ is *admissible* if:\n\n1. the restriction $\\pi|_K$ is unitary;\n2. for each [[irreducible representation|FX3NEFYFFN]] $\\tau$ of $K$, the multiplicity of $\\tau$ in $\\pi|_K$ is finite.\n\n## Remarks\n\nThe multiplicity of $\\tau$ is the dimension of $\\Hom_K(V_\\tau, V)$, the space of continuous $K$-equivariant linear maps from the space of $\\tau$ to $V$.\n\nExample: let $G = \\mathrm{SL}_2(\\R)$ and $K = \\mathrm{SO}(2)$, and let $G$ act on functions on $\\R^2 \\setminus \\set{0}$ that satisfy $f(tx) = \\abs{t}^{-1-\\nu} f(x)$ for $t \\ne 0$, with $\\nu \\in i\\R$, by $(\\pi(g) f)(x) = f(g^{-1} x)$.\nSuch an $f$ is determined by its values on the unit circle, and the norm is the $L^2$ norm there.\nThe characters of $K$ are $e^{i\\theta} \\mapsto e^{in\\theta}$ with $n \\in \\Z$.\nEach one with $n$ even occurs exactly once, spanned by the function $e^{in\\theta}$ on the circle, and the odd ones do not occur, so $\\pi$ is admissible.\n\nNon-example: the right regular representation of $G = \\mathrm{SL}_2(\\R)$ on $L^2(G)$ is not admissible.\nIts vectors fixed by $K$ are the functions invariant under right translation by $K$, which form the infinite-dimensional space $L^2(G/K)$.\nSo clause 2 fails for the trivial representation of $K$.\n\nOver a non-archimedean local field $F$, a representation of $G(F)$ on a complex vector space $V$ is admissible under two conditions.\nEvery vector is fixed by some compact open subgroup, and the space $V^{K'}$ is finite-dimensional for each compact open subgroup $K'$.\n",
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