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  "body": "\n## Summary\n\nUnitary induction builds a [[unitary representation|74BPYJNZTM]] of a [[locally compact group|AHV05ZBCYW]] $G$ from a unitary representation of a closed subgroup $H$.\nThe new space consists of square-integrable vector-valued functions on $G$ that transform under $H$ by the given representation, and $G$ acts by translation.\nIt extends the [[induced representation|JJP4JFH3C6]] of finite groups to subgroups of infinite index, such as lattices and parabolic subgroups.\nMackey's theory of induced representations uses it to build and classify irreducible representations, for example the principal series of semisimple Lie groups.\n\n## Explanations\n\n### L1 human\n\nSuppose you know how a closed subgroup $H$ of $G$ acts on a Hilbert space $K$.\nLook at functions on $G$ with values in $K$ whose behaviour along each coset $gH$ is dictated by the action of $H$.\nSuch a function is really a section of a bundle over the coset space $G/H$, and $G$ moves these sections around by translation.\nFor example, inducing the trivial representation of the trivial subgroup gives $L^2(G)$, where $G$ acts by left translation.\nInducing a character of $\\Z$ up to $\\R$ gives functions on the line that repeat up to a fixed phase factor after each step of length $1$.\n",
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