Extreme lattices and vexillar designs

We define a notion of vexillar design for the flag variety in the spirit of the already known spherical designs. We explain how the orbits of any flag under the action of a finite group can be a design. We show that a lattice is locally optimal for the general Hermite constant when its minima form a 4-design. The reasoning proves useful to show the extremality of many new expected examples (E8, Λ24, Barnes–Wall lattices, Thompson–Smith lattice for instance) that were out of reach until now.