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Modular representation theory in type A via Soergel bimodules

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.