Group invariant Colombeau generalized functions

Abstract

Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groups of arbitrary signature, such as groups of rotations or the Lorentz group. Further, a one-dimensional Colombeau generalized function with two (real) periods is shown to be a generalized constant, when the ratio of the periods is an algebraic nonrational number. Finally, a nonstandard Colombeau generalized function invariant under standard translations is shown to be constant.