Generalised θ operators on unitary Shimura varieties

The main result of this paper is the construction of a new class of weight shifting operators, similar to the theta operators of [16], [10] and others, which are defined on the lower Ekedahl-Oort strata of the geometric special fibre of unitary Shimura varieties of signature ( -1, 1) at a good prime , split in the in the reflex field , which we assume to be quadratic imaginary.These operators act on certain graded sheaves which are obtained from the arithmetic structure of the EO strata, in particular the -rank on each stratum.We expect these operators to have applications to the study of Hecke-eigensystems of (mod ) modular forms and generalisations of the weight part of Serre's conjecture.