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  "body": "\n## Definition\n\nLet $G$ be a second countable [[locally compact group|AHV05ZBCYW]], and let $H$ be a closed subgroup of $G$.\nAssume that the coset space $G/H$ carries a nonzero $G$-invariant Radon measure $\\nu$.\nLet $\\sigma$ be a [[unitary representation|74BPYJNZTM]] of $H$ on a [[Hilbert space|HM2HHE3J51]] $K$.\nLet $\\mathcal{H}$ be the space of measurable functions $f \\colon G \\to K$ such that:\n\n1. $f(gh) = \\sigma(h)^{-1} f(g)$ for all $g \\in G$ and $h \\in H$;\n2. $\\int_{G/H} \\norm{f(g)}^2 \\, d\\nu(gH) \u003c \\infty$.\n\nFunctions that agree almost everywhere are identified.\nThe norm $\\norm{f(g)}$ depends only on the [[coset|03DEH43F4H]] $gH$, because $\\sigma$ is unitary, so clause 2 makes sense.\nThe *induced representation* $\\operatorname{Ind}_H^G \\sigma$ is the representation of $G$ on $\\mathcal{H}$ given by $(\\lambda(x) f)(g) = f(x^{-1} g)$.\n\n## Remarks\n\nExample: take $H = \\set{e}$ and $\\sigma$ trivial on $K = \\C$.\nClause 1 is empty, $G/H = G$, and $\\nu$ is a left [[Haar measure|VY0HXD3GC4]].\nSo $\\operatorname{Ind}_{\\set{e}}^G 1$ is the left regular representation of $G$ on $L^2(G)$.\n\nExample: take $G = \\R$, $H = \\Z$, and $\\sigma(n) = e^{2 \\pi i \\theta n}$ for a fixed $\\theta \\in \\R$.\nClause 1 asks that $f(x + n) = e^{-2 \\pi i \\theta n} f(x)$, and clause 2 asks that $\\int_0^1 \\abs{f(x)}^2 \\, dx \u003c \\infty$.\n\nNon-example: in the second example, the constant function $f = 1$ fails clause 1 when $\\theta \\notin \\Z$.\n\nWhen $G/H$ has no invariant measure, one uses a quasi-invariant measure instead.\nThe action then includes the square root of a Radon–Nikodym factor, so that $\\lambda(x)$ stays unitary.\nFor finite index and finite groups, the result agrees with the [[induced representation|JJP4JFH3C6]] of finite group theory.\n",
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