Smoothness of Schubert varieties in affine Grassmannians

The geometry in the reduction of Shimura varieties, respectively moduli spaces of Drinfeld shtukas plays a central role in the Langlands program, and it is desirable to single out cases of smooth reduction. Recent works of Kisin, Pappas and Zhu reduce this question to so called Schubert varieties which are defined purely in terms of linear algebra, and thus easier to handle.

We consider Schubert varieties which are associated with a reductive group G over a Laurent series local field, and a special vertex in the Bruhat-Tits building. If G splits, a strikingly simple classification is given by a theorem of Evans and Mirković. If G does not split, the analogue of their theorem fails: there is a single surprising additional case of "exotic smoothness“. In my talk, I explain how to obtain a complete list of the smooth and rationally smooth Schubert varieties. This is joint work with Tom Haines from Maryland.