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  "body": "\n## Definition\n\nLet $G$ be a [[real reductive Lie group|42ZCE14AD2]] with Lie algebra $\\mathfrak g_0$, and let $\\mathfrak g = \\mathfrak g_0 \\otimes_\\R \\C$ be its complexified Lie algebra.\nLet $K$ be a [[maximal compact subgroup|T2EB8R9HEV]] of $G$, with Lie algebra $\\mathfrak k_0 \\subseteq \\mathfrak g_0$, acting on $\\mathfrak g$ by the [[adjoint representation|P6ZP0TE07M]] $\\mathrm{Ad}$.\nA $(\\mathfrak g, K)$-module is a complex vector space $V$ with a representation of $\\mathfrak g$, written $X \\cdot v$, and a representation $\\pi$ of $K$, such that:\n\n1. each $v \\in V$ lies in a finite-dimensional $\\pi(K)$-invariant subspace on which $K$ acts continuously;\n2. $\\frac{d}{dt}\\big|_{t=0} \\pi(\\exp tX) v = X \\cdot v$ for all $X \\in \\mathfrak k_0$ and $v \\in V$;\n3. $\\pi(k)(X \\cdot v) = (\\mathrm{Ad}(k) X) \\cdot (\\pi(k) v)$ for all $k \\in K$, $X \\in \\mathfrak g$ and $v \\in V$.\n\nA *Harish-Chandra module* is a $(\\mathfrak g, K)$-module $V$ such that:\n\n4. $V$ is finitely generated as a module over the [[universal enveloping algebra|SCZ8PRVY00]] $U(\\mathfrak g)$;\n5. each [[irreducible representation|FX3NEFYFFN]] of $K$ occurs in $V$ with finite multiplicity.\n\n## Remarks\n\nThe derivative in clause 2 makes sense by clause 1, since it is taken inside a finite-dimensional space.\n\nExample: let $\\pi$ be a finite-dimensional continuous representation of $G$ on $V$, with differential $d\\pi$ extended complex-linearly to $\\mathfrak g$.\nThen $V$ with $X \\cdot v = d\\pi(X) v$ and $\\pi|_K$ is a Harish-Chandra module.\nClause 1 holds because $V$ itself is finite-dimensional, clauses 2 and 3 hold by the definition of $d\\pi$ and by $\\pi(k) \\pi(\\exp tX) \\pi(k)^{-1} = \\pi(\\exp t \\mathrm{Ad}(k) X)$, and clauses 4 and 5 hold because $\\dim V \u003c \\infty$.\n\nNon-example: an infinite direct sum of copies of the trivial one-dimensional module, with $\\mathfrak g$ acting by $0$ and each $k \\in K$ acting as the identity, is a $(\\mathfrak g, K)$-module.\nIt is not a Harish-Chandra module, because the trivial representation of $K$ occurs infinitely often and clause 5 fails.\n\nIf $\\pi$ is an [[admissible representation|SZV8Y806Y7]] of $G$ on a Hilbert space $H$, its $K$-finite vectors are smooth and form a Harish-Chandra module when $\\pi$ has finite length.\nTwo such representations are called infinitesimally equivalent when their Harish-Chandra modules are isomorphic.\n",
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