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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]] with a left [[Haar measure|VY0HXD3GC4]] $\\mu$, and let $K$ be a [[compact|0R6EWFKRCF]] [[open subgroup|QYCFH68RD1]] of $G$.\nThe *Hecke algebra* $\\mathcal{H}(G, K)$ is the set of functions $f \\colon G \\to \\C$ such that:\n\n1. $f(k_1 g k_2) = f(g)$ for all $g \\in G$ and $k_1, k_2 \\in K$;\n2. $f$ has compact support.\n\nIts product is [[convolution|2Z29JN3VES]]:\n$$(f_1 * f_2)(x) = \\int_G f_1(y) f_2(y^{-1} x) \\, d\\mu(y).$$\n\n## Remarks\n\nEach double coset $KgK$ is compact and open, and a compact set meets only finitely many of them.\nSo the indicator functions $\\mathbf{1}_{KgK}$ form a basis of $\\mathcal{H}(G, K)$ as a vector space.\nThe function $\\mu(K)^{-1} \\mathbf{1}_K$ is the unit element.\n\nExample: $G$ discrete and $K = \\set{e}$, with $\\mu$ the counting measure.\nBoth clauses say only that $f$ is finitely supported, and $\\mathbf{1}_{\\set{a}} * \\mathbf{1}_{\\set{b}} = \\mathbf{1}_{\\set{ab}}$.\nSo $\\mathcal{H}(G, \\set{e})$ is the group algebra $\\C[G]$.\n\nExample: $G = \\mathrm{GL}_2(\\Q_p)$ and $K = \\mathrm{GL}_2(\\Z_p)$.\nThen $\\mathcal{H}(G, K)$ is commutative.\nIt is generated by the indicators of $K \\operatorname{diag}(p, 1) K$ and $K \\operatorname{diag}(p, p) K$ and the inverse of the latter.\n\nNon-example: for $G = \\Q_p$ and $K = \\Z_p$, the indicator of $p \\Z_p$ fails clause 1, since it is not constant on the coset $\\Z_p$.\n\nWhen $G$ is totally disconnected, the union of the $\\mathcal{H}(G, K)$ over all compact open $K$ is the full Hecke algebra $\\mathcal{H}(G)$ of locally constant compactly supported functions.\n",
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