Special cycles on unitary Shimura varieties I. Unramified local theory

Abstract

The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n−1) over ℚ is uniformized by a formal scheme $\mathcal{N}$ . In the case when p is an inert prime, we define special cycles ${\mathcal{Z}}({\bold x})$ in $\mathcal{N}$ , associated to collections ${\bold x}$ of m ‘special homomorphisms’ with fundamental matrix T∈Herm m (O k ). When m=n and T is nonsingular, we show that the cycle ${\mathcal{Z}}({\bold x})$ is either empty or is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient condition, in terms of T, for ${\mathcal{Z}}({\bold x})$ to be irreducible. When ${\mathcal{Z}}({\bold x})$ is zero dimensional—in which case it reduces to a single point—we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.