Extremality and designs in spaces of quadratic forms

A well known theorem of Voronoi caracterizes extreme quadratic forms and Euclidean lattices, that is those which are local maxima for the Hermite function, as perfect and eutactic. This caracterization has been extended in various cases, such that family of lattices, sections of lattices, Humbert forms, etc. Moreover, there is a criterion for extreme lattices, discovered by Venkov, formulated in terms of spherical designs which has been extended in the case of Grassmannians and sections of lattices. In this article, we define a general frame, in which there is a Voronoi characterization'', and a Venkov criterion’’ through an appropriate notion of design. This frame encompasses many interesting situations in which a ``Voronoi characterization’’ has been proved. We also discuss the question of extremality relatively to the Epstein zeta function, and we extend to our frame a characterization of final zeta-extremality formulated by Delone and Ryshkov and a criterion in terms of designs found by Coulangeon.