Schur-Weyl duality and categorification

In some joint work with Kleshchev in 2008, we discovered a higher level analog of Schur-Weyl duality, relating parabolic category O for the general linear Lie algebra to certain cyclotomic Hecke algebras. Meanwhile Rouquier and others were developing a general axiomatic approach to the study of categorical actions of Lie algebras. In this survey, we recall aspects of these two theories, then explain some related recent developments due to Losev and Webster involving tensor product categorifications.