Orbital integral

In mathematics, an orbital integral is an integral that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres: for a homogeneous space X = G/H, a generalized sphere centered at a point x₀ is a group orbit of the isotropy group of x₀. The model case for orbital integrals is a symmetric space G/K where G is a Lie group and K a necessarily compact unimodular subgroup. Generalized spheres in this case are then actual geodesic spheres, and one can define the spherical averaging operator directly by

$M^{r}f(x)=\int _{K}f(gk\cdot y)\,dk$

where

• the dot product denotes the action of the group G on the homogeneous space X
• g ∈ G is a group element such that x = g·o
• y ∈ X is an arbitrary element of the geodesic sphere of radius r centered at x: d(x,y) = r
• the integration is taken with respect to the Haar measure on K.

A central problem of integral geometry is to reconstruct a function from knowledge of its orbital integrals. The Funk transform and Radon transform are two special cases. In the theory of automorphic forms, orbital integrals enter naturally into the formulation of the stable trace formula.

References

• Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, ISBN 0-12-338301-3