Affine Grassmannians of group schemes and exotic principal bundles over ${\Bbb A}^1$

Let ${\bf G}$ be a simple simply-connected group scheme over a regular local scheme $U$. Let ${\mathcal E}$ be a principal ${\bf G}$-bundle over ${\Bbb A}^1_U$ trivial away from a subscheme finite over $U$. We show that ${\mathcal E}$ is not necessarily trivial and give some criteria of triviality. To this end, we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal ${\bf G}$-bundles over ${\Bbb A}^1_U$ with split ${\bf G}$ such that the bundles are not isomorphic to pullbacks from $U$.