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A dimension formula for Ekedahl-Oort strata. (English) Zbl 1062.14033
To an abelian variety \(X\) over a field \(k\) of characteristic \(p\) one can associate invariants such as the \(p\)-rank or the isogeny class of its \(p\)-divisible group, and such invariants determine stratifications of the moduli space \(\mathcal A_g\). The Ekedahl-Oort stratification is defined by declaring that two points of \(\mathcal A_g (k)\) are in the same stratum if and only if the associated group schemes \(X[p]\) are isomorphic over \(\overline{k}\). Such stratifications can be extended to other Shimura varieties.
Let \(G\) be a connected reductive group associated to the given moduli problem, and let \(\mathbb X\) be a conjugacy class of parabolic subgroups of \(G\). Let \(W_G\) be the Weyl group of \(G\), and let \(W_\mathbb X\) be the subgroup of \(W_G\) associated to \(\mathbb X\). Then there is a generalized Ekedahl-Oort stratification on good reductions \(\mathcal A_0\) of PEL moduli spaces of the form \(\mathcal A_0 = \coprod_{w \in W_\mathbb X \backslash W_G} \mathcal A_0 (w)\). Note that, given a fixed set \(S \subset W_G\) of reflections, each coset \(w \in W_\mathbb X \backslash W_G\) has a distinguished representative \(\dot{w}\).
In this paper the author proves that the irreducible components of \(\mathcal A_0 (w)\) all have dimension equal to the length \(\ell (\dot{w})\) of \(\dot{w}\) if \(\mathcal A_0 (w) \neq 0\).

14G35 Modular and Shimura varieties
11G15 Complex multiplication and moduli of abelian varieties
14L15 Group schemes
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