Affine Grassmannians of group schemes and exotic principal bundles over A¹

Let G be a simple simply-connected group scheme over a regular local scheme U . Let E be a principal G-bundle over A 1 U trivial away from a subscheme finite over U . We show that 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 G-bundles over A 1 U with split G such that the bundles are not isomorphic to pullbacks from U .