Gelfand pair

Summary

A Gelfand pair is a locally compact group $G$ with a compact subgroup $K$ such that convolution of $K$-bi-invariant functions on $G$ is commutative. Equivalently, each irreducible unitary representation of $G$ has at most a one-dimensional space of $K$-fixed vectors. For a Gelfand pair, harmonic analysis on the homogeneous space $G/K$ is governed by spherical functions. They play the role of the exponentials in Fourier analysis. Examples are $(\mathrm{SO}(3), \mathrm{SO}(2))$, related to the sphere, and $(\mathrm{SL}_2(\R), \mathrm{SO}(2))$, related to the hyperbolic plane.