Spherical Functions on Euclidean Space

Details Spherical Functions on Euclidean Space Spherical Functions on Euclidean Space

2005-09-20

arxiv.org/abs/math/0509459v1

Mathematics

Representation Theory

10 pages

Scientific paper

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space $E^n = G/K$ where $G$ is the semidirect product $R^n \cdot K$ of the translation group with a closed subgroup $K$ of the orthogonal group O(n). We give exact parameterizations of the space of $(G,K)$--spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where $K$ is transitive on the unit sphere in $E^n$.

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