Extreme lattices and vexillar designs

We define a notion of vexillar design for the flag variety in the spirit of the spherical designs introduced by Delsarte, Goethals and Seidel. For a finite subgroup of the orthogonal group, we explain how conditions on the group have the orbits of any flag under the group action be a design and point out why the minima of a lattice in the sense of the general Hermite constant forming a 4-design implies being extreme. The reasoning proves useful to show the extremality of many new expected examples ($E_8$, $\La_{24}$, Barnes-Wall lattices, Thompson-Smith lattice for instance) that were out of reach until now.