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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
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is compact (but not necessarily symmetric), a similar shortcut works. The problem is more interesting when
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Groups and
Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions
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is non-compact. For example, the Radon transform is the orbital integral that results by taking
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and the left and right Haar measures coincide and can be normalized so that the mass of
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Orbital integrals of suitable functions can also be defined on homogeneous spaces
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is to reconstruct a function from knowledge of its orbital integrals. The
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Orbital integrals are an important technical tool in the theory of
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563:, where they enter into the formulation of various
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410:is a group element that represents the coset
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530:{\displaystyle f(x)=\lim _{r\to 0^{+}}M^{r}f(x).}
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93:The model case for orbital integrals is a
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388:{\displaystyle \int _{K}f(gk\cdot y)\,dk}
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220:the dot denotes the action of the group
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16:Integral transform type in mathematics
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556:the isotropy group of a hyperplane.
398:is the orbital integral centered at
333:with respect to the Haar measure of
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224:on the homogeneous space
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63:centered at a point
316:where the subgroup
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578:Helgason, Sigurdur
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296:is compact, it is
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61:generalized sphere
37:homogeneous spaces
29:integral transform
682:Automorphic forms
672:Harmonic analysis
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561:automorphic forms
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418:Integral geometry
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21:mathematics
666:Categories
571:References
325:-orbit in
298:unimodular
89:Definition
488:→
370:⋅
349:∫
185:⋅
164:∫
110:Lie group
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