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  "body": "\n## Definition\n\nLet $G$ be a connected semisimple Lie group with finite center that is a closed [[subgroup|0X1JXT47GA]] of $\\mathrm{GL}_n(\\R)$.\nLet $K$ be a maximal compact subgroup of $G$, and let $\\mathfrak{g}$ be the complexified Lie algebra of $G$.\nLet $Z(\\mathfrak{g})$ be the center of its universal enveloping algebra.\nIt acts on smooth functions on $G$ as left-invariant differential operators.\nLet $\\Gamma$ be a discrete subgroup of $G$, in the main case an arithmetic subgroup.\nFor $g \\in G$ let $\\norm{g} = \\max_{i,j} \\max(\\abs{g_{ij}}, \\abs{(g^{-1})_{ij}})$.\nAn *automorphic form* for $\\Gamma$ is a [[smooth function|0V33BNKZ9S]] $f \\colon G \\to \\C$ such that:\n\n1. ($\\Gamma$-invariance) $f(\\gamma g) = f(g)$ for all $\\gamma \\in \\Gamma$ and $g \\in G$.\n2. ($K$-finiteness) The functions $g \\mapsto f(gk)$ for $k \\in K$ span a finite-dimensional vector space.\n3. ($Z(\\mathfrak{g})$-finiteness) There is an ideal $J$ of finite codimension in $Z(\\mathfrak{g})$ with $Df = 0$ for all $D \\in J$.\n4. (Moderate growth) There are constants $C, N \u003e 0$ with $\\abs{f(g)} \\le C \\norm{g}^N$ for all $g \\in G$.\n\n## Remarks\n\nExample: let $G = \\mathrm{SL}_2(\\R)$, $K = \\mathrm{SO}(2)$, $\\Gamma = \\mathrm{SL}_2(\\Z)$, and let $h$ be a [[modular form|GWKN5VG78A]] of weight $k$.\nFor $g = \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ put $f(g) = h(g \\cdot i)\\,(ci + d)^{-k}$, where $g \\cdot i = \\frac{ai + b}{ci + d}$.\nThe modularity of $h$ gives clause 1.\nRight translation by the rotation $k_\\theta = \\begin{pmatrix} \\cos\\theta \u0026 -\\sin\\theta \\\\ \\sin\\theta \u0026 \\cos\\theta \\end{pmatrix}$ multiplies $f$ by $e^{-ik\\theta}$, since $k_\\theta \\cdot i = i$ and $i \\sin\\theta + \\cos\\theta = e^{i\\theta}$, which gives clause 2.\nHolomorphy of $h$ makes the Casimir operator act on $f$ by a scalar, which gives clause 3.\nThe bound of $h$ at the cusp gives clause 4.\n\nNon-example: on the same $G$, the function $f(g) = (\\mathrm{Im}\\, g \\cdot i)^s$ satisfies clauses 2, 3 and 4, but it is not invariant under $\\gamma = \\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix} \\in \\Gamma$, so clause 1 fails.\n\nThis is the definition of Harish-Chandra.\nA classical variant works on a symmetric space $G/K$ and allows a factor of automorphy $j(\\gamma, z)$ in clause 1, as for modular forms of weight $k$.\nIn the adelic setting, $G$ is a reductive group over a number field $F$ and automorphic forms are functions on $G(F) \\backslash G(\\mathbb{A}_F)$.\n",
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