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  "body": "\n## Summary\n\nA principal series representation of a real reductive group is induced from a minimal parabolic subgroup.\nThe inducing data are a character of its split part and an irreducible representation of its compact part.\nFor $\\mathrm{SL}_2(\\R)$, it acts on homogeneous functions on the plane, or equivalently on functions on the circle.\nThe unitary principal series supply the continuous part of the Plancherel decomposition of $L^2(G)$.\nBy a theorem of Casselman, every irreducible admissible representation embeds into some principal series, which makes these representations a universal source.\n\n## Explanations\n\n### L1 human\n\nPick a large subgroup $P$ of $G$, such as the upper triangular matrices in $\\mathrm{SL}_2(\\R)$.\nA character of $P$ is a simple one-dimensional rule.\nInducing spreads this rule over the whole group: the space consists of functions on $G$ that transform under $P$ by the chosen character.\nThe resulting family depends on a complex parameter $\\nu$, and almost all its members are irreducible.\n",
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