Harmonic analysis on homogeneous spaces

Summary

Harmonic analysis on a homogeneous space $G / H$ studies functions on $G / H$ through the action of the group $G$. The central question is how the representation of $G$ on $L^2(G / H)$ decomposes into irreducible unitary representations. For $H$ trivial this is harmonic analysis on the group $G$ itself. For $H$ compact with $(G, H)$ a Gelfand pair, it leads to spherical functions and the spherical transform. Classical examples are Fourier series on the circle, spherical harmonics on the sphere and analysis on the hyperbolic plane.