Transformationsgruppen mit affinen Bahnen. (Transformation groups with affine orbits)
Let G be a connected Lie group acting on \({\mathbb{R}}^ n\) in such a manner that all orbits are affine and that the Euclidean measures on the orbits are G-invariant. In this paper there are given explicit disintegrations of invariant measures and distributions, respectively, over the set of Euclidean measures. In the case of linear group actions the Euclidean measures are closely related with invariant eigendistributions of invariant differential operators with constant coefficients. Thus the disintegrations provide eigendistribution expansions. The eigendistribution expansions on Heisenberg groups given by \textit{L. P. Rothschild} and \textit{J. A. Wolf} [Indiana Univ. Math. J. 25, 753-762 (1976; Zbl 0353.22007)] are received as a special case. Furthermore, from our disintegration formula we derive necessary and sufficient conditions for certain invariant differential equations to have invariant solutions.
invariant eigendistributions of invariant differential operators
scientific article; zbMATH DE number 3932068