On modular categories O for quantized symplectic resolutions

In this paper we study highest weight and standardly stratified structures on modular analogs of categories O over quantizations of symplectic resolutions. We also show how to recover the usual categories O (reduced mod p >> 0) from our modular categories. More precisely, we consider a conical symplectic resolution that is defined over a finite localization of Z and is equipped with a Hamiltonian action of a torus T that has finitely many fixed points. We consider algebras A(lambda) of global sections of a quantization in characteristic p >> 0, where lambda is a parameter. Then we consider a category (O) over tilde (lambda) consisting of all finite dimensional T -equivariant A(lambda)-modules. Under reasonable assumptions that hold in most examples of interest, we show that for lambda lying in a p-alcove (p)A, the category (O) over tilde (lambda) is highest weight (in some generalized sense). Moreover, we show that every face of (p)A that survives in (p)A/p when p -> infinity defines a standardly stratified structure on (O) over tilde (lambda) We identify the associated graded categories for these standardly stratified structures with reductions mod p of the usual categories O in characteristic 0. Applications of our construction include computations of wall-crossing bijections in characteristic p and the existence of gradings on categories O in characteristic 0.