The model case for orbital integrals is a Riemannian symmetric space G/K, where G is a Lie group and K is a symmetric compact subgroup.
Generalized spheres are then actual geodesic spheres and the spherical averaging operator is defined as where Orbital integrals of suitable functions can also be defined on homogeneous spaces G/K where the subgroup K is no longer assumed to be compact, but instead is assumed to be only unimodular.
When G/K is a Riemannian symmetric space, the problem is trivial, since Mrƒ(x) is the average value of ƒ over the generalized sphere of radius r, and When K is compact (but not necessarily symmetric), a similar shortcut works.
For example, the Radon transform is the orbital integral that results by taking G to be the Euclidean isometry group and K the isotropy group of a hyperplane.
Orbital integrals are an important technical tool in the theory of automorphic forms, where they enter into the formulation of various trace formulas.