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  "body": "\n## Statement\n\nLet $G$ be a linear connected [[real reductive Lie group|42ZCE14AD2]], and fix a minimal parabolic subgroup $P_0$ of $G$.\nConsider the triples $(P, \\sigma, \\nu)$ such that:\n\n1. $P \\supseteq P_0$ is a [[parabolic subgroup|Z2FEBW5MQM]] with [[Langlands decomposition|BM3TWJ0D4P]] $P = MAN$;\n2. $\\sigma$ is an equivalence class of irreducible tempered representations of $M$;\n3. $\\nu \\in \\mathfrak a^*_\\C$, where $\\mathfrak a$ is the Lie algebra of $A$, and $\\operatorname{Re} \\nu$ is strictly positive on every root of $\\mathfrak a$ in the Lie algebra of $N$.\n\nFor such a triple, the normalised [[induced representation|6Q7AHAZ6Q4]] $\\mathrm{Ind}_P^G(\\sigma \\otimes e^\\nu \\otimes 1)$ has a unique irreducible quotient $J(P, \\sigma, \\nu)$, the Langlands quotient.\nEvery irreducible [[admissible representation|SZV8Y806Y7]] of $G$ is infinitesimally equivalent to exactly one $J(P, \\sigma, \\nu)$.\n\n## Remarks\n\nInfinitesimal equivalence means that the [[Harish-Chandra modules|2MKHH636C3]] are isomorphic.\nThe proof compares the leading exponents of the matrix coefficients of a representation with the parameter $\\nu$.\nThe quotient $J(P, \\sigma, \\nu)$ is the image of a standard intertwining operator, which converges because $\\operatorname{Re} \\nu$ lies in the positive chamber.\nIf $G$ is semisimple and $P = G$, then $A$ is trivial, so $\\nu = 0$, and $J(G, \\sigma, 0) = \\sigma$ runs over the tempered representations themselves.\nExample: let $G = \\mathrm{SL}_2(\\R)$ and let $P$ be the upper triangular matrices.\nThe constant functions form a subrepresentation of the [[principal series|CWDF4KGHS3]] $I(\\mathrm{triv}, -\\rho)$, and dually the trivial representation is a quotient of $I(\\mathrm{triv}, \\rho)$.\nSo $J(P, \\mathrm{triv}, \\rho)$ is the trivial representation.\nThe positivity of $\\operatorname{Re} \\nu$ is needed: on the unitary axis $\\operatorname{Re} \\nu = 0$ the induced representation can be reducible with several irreducible quotients, so uniqueness fails.\n\n## Explanations\n\n### L1 human\n\nThe theorem reduces the classification of all irreducible representations to that of the tempered ones, which are the representations that appear in $L^2(G)$.\nEach irreducible representation is built from a tempered representation of a smaller group and a parameter that measures its growth.\n\n### L2 human\n\nLanglands proved the classification for real groups; the analogous result for $p$-adic groups is due to Borel–Wallach and Silberger.\nTempered representations are in turn built from discrete series of Levi subgroups, so the whole picture rests on Harish-Chandra's discrete series.\nThe classification is the starting point of the local Langlands correspondence for real groups, which attaches $L$-parameters to the triples $(P, \\sigma, \\nu)$.\n",
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