Minimal rational curves on wonderful group compactifications
Let G be a simple algebraic group of adjoint type, and V an irreducible projective representation. The closure of the image of G in the projectivization of End(V) is an equivariant compactification X(V) of the group G. We answer the following questions: Is X(V) covered by lines? If not, what is the minimal degree of a family of rational curves covering X(V), and what is the structure of such a family? In particular, one may find V such that X(V) is covered by lines, except when G has type E8. The combinatorics of the highest short coroot play an important role in these questions. This is a joint work with Michel Brion.