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  "body": "\n## Summary\n\nA Gelfand pair is a [[locally compact group|AHV05ZBCYW]] $G$ with a compact subgroup $K$ such that convolution of $K$-bi-invariant functions on $G$ is commutative.\nEquivalently, each irreducible unitary representation of $G$ has at most a one-dimensional space of $K$-fixed vectors.\nFor a Gelfand pair, harmonic analysis on the homogeneous space $G/K$ is governed by [[spherical functions|BKZNC76RC8]].\nThey play the role of the exponentials in Fourier analysis.\nExamples are $(\\mathrm{SO}(3), \\mathrm{SO}(2))$, related to the sphere, and $(\\mathrm{SL}_2(\\R), \\mathrm{SO}(2))$, related to the hyperbolic plane.\n\n## Explanations\n\n### L1 human\n\nThink of functions on the sphere that only depend on the latitude.\nThey can be seen as functions on the rotation group that do not change under rotations around the north pole, on either side.\nFor such functions, convolution does not depend on the order of the factors.\nThis commutativity makes their analysis almost as simple as Fourier series.\nA group with a subgroup that behaves like this is called a Gelfand pair.\nThe pair of all rotations and the rotations around one axis is the main example.\n",
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