G-adapted deformations and Ekedahl-Oort stratification of Shimura varieties

2017.01.03（星期二），14:00-15:00

点：数学院南楼N818

Abstract:

Ekedahl-Oort stratification was firstly defined and studied by Oort for the moduli space $\mathcal{A}_{g, \mathbb{F}_p}$ of principally polarized abelian varieties over $\mathbb{F}_p$. This notion has been generalized and studied by Moonen, Wedhorn, Viehmann and Zhang for good reduction of general Shimura varieties of Hodge type. Let $W$ be the Weyl group of the reductive group $G$ of a mod $p$ Shimura variety $S$.  Then the Ekedahl-Oort strata of $S$ are parametrized by a certain subset ${}^JW$ of $W$. In one of the recent works of Viehmann, it is showed that ${}^JW$  corresponds naturally to some objects coming from the loop group $\mathcal{L}G$ of $G$.  But this correspondence is purely group theoretic and hence one naturally asks the question: is it possible to give a direct connection between $S$ and  $\mathcal{L}G$ (the latter is an important object from both the geometric and the arithmetic points of views)?  In this talk I will explain that this connection is indeed possible. To give the connection we use the classification result of $p$-divisible groups in term of Breuil-Kisin modules (equivalently Breuil-Kisin windows) and $G$-adapted deformations (a notion we borrow from Kisin).