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Abstract We give a simple and uniform proof of a conjecture of Haines–Richarz characterizing the smooth locus of Schubert varieties in twisted affine Grassmannians. Our method is elementary and avoids any representation theoretic techniques, instead relying on a combinatorial analysis of tangent spaces of Schubert varieties.
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We show how to characterize integral models of Shimura varieties over places of the reflex field where the level subgroup is parahoric by formulating a definition of a ``canonical" integral model. We then prove that, in Hodge type cases and under a tameness hypothesis, the integral models constructed by the author and Kisin in previous work are canonical and, in particular, independent of choices. A main tool is a theory of displays with parahoric structure that we develop in this paper.more » « less
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We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular $$p$$ -adic integral models for these varieties over odd primes $$p$$ at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model.more » « less
