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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 $$\mathcal A_0 (w) \neq 0$$.

##### MSC:
 14G35 Modular and Shimura varieties 11G15 Complex multiplication and moduli of abelian varieties 14L15 Group schemes
##### Keywords:
abelian varieties; Shimura varieties; finite group schemes
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##### References:
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