Kempf-Ness covariant and reduction theory

Let Γ be a group acting on a set X. In order to facilitate the study of the quotient Γ \ X, one is often led to choose, in each Γ-orbit, a distinguished representative, which is said to be reduced. The set of reduced points then forms a fundamental domain for the action of Γ on X. If one wishes to perform computations in Γ \ X, it is desirable that the fundamental domain admit an explicit description, and that the reduced representative of a point in X can be determined algorithmically. One then says that a reduction theory for the action of Γ on X has been defined. I will show how to solve this problem in a very general setting of great importance in number theory, by means of a geometric object: the Kempf–Ness covariant.