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  "body": "\n## Statement\n\nLet $G$ be a connected [[real reductive Lie group|42ZCE14AD2]] with finite center and Haar measure $dg$.\nLet $G'$ be the open dense subset of regular semisimple elements of $G$.\nLet $\\mathfrak Z$ be the center of the [[universal enveloping algebra|SCZ8PRVY00]] of the complexified Lie algebra of $G$.\nIt acts on distributions on $G$ as bi-invariant differential operators.\nLet $\\Theta$ be a distribution on $G$ such that:\n\n1. $\\Theta$ is invariant under conjugation by $G$;\n2. there is a homomorphism $\\chi \\colon \\mathfrak Z \\to \\C$ with $z \\Theta = \\chi(z) \\Theta$ for all $z \\in \\mathfrak Z$.\n\nThen there is a locally integrable function $F$ on $G$, real analytic on $G'$ and constant on [[conjugacy classes|0DEBXAJ5SA]], with the following property.\nFor every smooth compactly supported $f$ on $G$, one has $\\Theta(f) = \\int_G f(g) F(g) \\, dg$.\nIn particular, the [[Harish-Chandra character|F2E3317P78]] $\\Theta_\\pi$ of an irreducible [[admissible representation|SZV8Y806Y7]] $\\pi$ is given by such a function.\n\n## Remarks\n\nThe character of an irreducible admissible representation satisfies the hypotheses, because $\\mathfrak Z$ acts on such a representation by scalars, its infinitesimal character.\nThe proof first treats invariant eigendistributions on the Lie algebra, using the radial parts of invariant differential operators along Cartan subalgebras.\nIt then transfers the result to the group through the exponential map near each semisimple element.\nOn $G'$ the eigen-equations become an elliptic system, which gives analyticity there; the hard part is the local integrability across the singular set $G \\setminus G'$.\nFor a finite-dimensional representation, $F$ is the ordinary character $g \\mapsto \\operatorname{tr} \\pi(g)$, which is analytic on all of $G$.\nFor infinite-dimensional representations $F$ is usually singular on $G \\setminus G'$, so analyticity on all of $G$ cannot be expected.\n\n## Explanations\n\n### L1 human\n\nThe character of an infinite-dimensional representation is a priori only a generalised function.\nThe theorem says that it is in fact an honest function, smooth away from a thin bad set, and still integrable near that set.\nThis makes it possible to write characters by explicit formulas, as for finite groups.\n\n### L2 human\n\nThe theorem is the starting point of Harish-Chandra's harmonic analysis on real reductive groups.\nExplicit formulas for the function $F$ on each Cartan subgroup lead to the construction of the discrete series and to the Plancherel formula.\nAn analogue holds for reductive groups over $p$-adic fields, also due to Harish-Chandra, where the proof is different.\n",
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