Ekedahl-Oort strata with a superspecial factor in unitary Shimura varieties

Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic $p>2$ and let $O_K$ be an imaginary quadratic order in which $p$ is inert. For any $a,b\in \mathbb{Z}_{\geq 0}$ not both 0, denote by $M(a,b)$ the unitary Shimura variety that parametrizes principally polarized abelian varieties over $\mathbb{k}$ with an action of signature $(a,b)$ by $O_K$ (subject to certain compatibility conditions). For any $a,b,c,d \in \mathbb{Z}_{\geq 0}$, there is a natural product map $M(a,b) \times M(c,d) \to M(a+c,b+d)$, which induces a map on Ekedahl-Oort strata. In this paper we focus on product maps where at least one of the factors is superspecial, since most Ekedahl-Oort strata of unitary Shimura varieties contain such a factor. We describe maps with this property completely on the level of Weyl group coset representatives. We also characterize, on the level of Weyl group coset representatives, which Ekedahl-Oort strata of $M(a,b)$ admit a superspecial factor.