Title:
Integral affine Graßmannians of twisted groups and local models of Shimura varieties
Abstract:
Local models of Shimura varieties are integral models of flag varieties which help in understanding the local geometric behaviour of arithmetic models of Shimura varieties and were first systematically introduced by Rapoport-Zink in EL and PEL cases. More recently, a group-theoretic approach to their definition and study has been possibilitated by the theory of affine Graßmannians, as in the works of Pappas-Rapoport and Pappas-Zhu, where the authors always assume tame ramification. We generalise the constructions of these last papers, by exhibiting certain smooth affine and connected “parahoric” group models over Z[t] of a given quasi-split Q(t)-group G with absolutely simple simply connected cover splitting over the normal closure of Q(t^{1/e}) with e=2 or 3 (under a mild assumption on the maximal torus). In characteristic e, the group scheme becomes generically pseudo-reductive and we explain in which sense the F_e[t]-model may still be interpreted as parahoric. Then we focus on the affine Graßmannians (both local and global) attached to this group scheme, which are proved to be representable by an ind-projective ind-scheme. We also obtain normality theorems for Schubert varieties in the local and global case (except if G is an odd dimensional unitary group) and an enumeration of the irreducible components of the fibres via the admissible set. Time permitting, we will explain how in the abelian case these global Schubert varieties give rise to the local models conjectured by Scholze.
This is a one-off seminar at non-standard time and date and place.