In mathematics, an orbital integral is an

integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...

that generalizes the spherical mean operator to

homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ...

s. Instead of integrating over

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

s, one integrates over generalized spheres: for a homogeneous space ''X'' = ''G''/''H'', a generalized sphere centered at a point ''x''₀ is an

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...

of the isotropy group of ''x''₀.

Definition

The model case for orbital integrals is a

Riemannian symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...

''G''/''K'', where ''G'' is a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

and ''K'' is a symmetric

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

. Generalized spheres are then actual

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

spheres and the spherical averaging operator is defined as :

M^rf(x) = \int_K f(gk\cdot y)\,dk,

where * the dot 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'' (since ''K'' is compact, it is unimodular and the left and right Haar measures coincide and can be normalized so that the mass of ''K'' is 1). 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. Lorentzian symmetric spaces are of this kind. The orbital integrals in this case are also obtained by integrating over a ''K''-orbit in ''G''/''K'' with respect to the Haar measure of ''K''. Thus :

\int_K f(gk\cdot y)\,dk

is the orbital integral centered at ''x'' over the orbit through ''y''. As above, ''g'' is a group element that represents the coset ''x''.

Integral geometry

A central problem of

integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformat ...

is to reconstruct a function from knowledge of its orbital integrals. The Funk transform and

Radon transform In mathematics, the Radon transform is the integral transform which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the ...

are two special cases. When ''G''/''K'' is a Riemannian symmetric space, the problem is trivial, since ''M''^''r''ƒ(''x'') is the average value of ƒ over the generalized sphere of radius ''r'', and :

f(x) = \lim_ M^rf(x).

When ''K'' is compact (but not necessarily symmetric), a similar shortcut works. The problem is more interesting when ''K'' is non-compact. 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 In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

, where they enter into the formulation of various trace formulas.

References

* Harmonic analysis {{mathanalysis-stub