Split subvarieties of group embeddings

Let G be a connected reductive group and X an equivariant compactification of G. In X, we study generalised and opposite generalised Schubert varieties, and their intersections called generalised Richardson varieties and projected generalised Richardson varieties. Any complete G-embedding has a canonical Frobenius splitting, and we prove that the compatibly split subvarieties are the generalised projected Richardson varieties extending a result of Knutson, Lam and Speyer to the situation.