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

Speaker: Qijun Yan ( Universiteit Leiden & Università degli Studi di Milano)

Time & Place: 2017-01-03, 14:00-15:00,  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).