Standard monomials for wonderful group compactifications

Let X be the wonderful compactification of the semisimple adjoint algebraic group G. We show that the basis of H0(X,L) constructed by Chirivi` and Maffei is compatible with all B×B-orbit closures in X by defining subsets using only combinatorics of the underlying paths. Furthermore, we construct standard monomials on X that have properties similar to classical standard monomials.