  In this paper we express certain multiplicities in modular
representation-theoretic categories of type A in terms of affine
p-Kazhdan-Lusztig polynomials. The representation-theoretic categories we deal
with include the categories of rational representations of GL(n),
representations of the quantum group for gl(n), and representations of
(degenerate) cyclotomic Hecke and Schur algebras, where the base field is an
algebraically closed field of arbitrary prime characteristic. In order to
approach this problem we define Soergel-theoretic versions of parabolic
categories O in characteristic p. We show that these categories have many
common features with the classical parabolic categories O; for example, they
are highest weight. We produce a homomorphism from a (finite or affine) type A
2-Kac-Moody category to the diagrammatic version of the category of singular
Soergel bimodules (again, of finite or affine type A). This leads to a
categorical Kac-Moody action on the Soergel-theoretic categories O. Then we
relate the representation-theoretic categories to Soergel-theoretic ones by
proving a uniqueness result for highest weight categorical actions on Fock
spaces.
