Let Gn=GLn(F), where F is a non-archimedean local field with residue characteristic p. Our starting point is the Bernstein decomposition of the representation category of Gn over an algebraically closed field of positive characteristic ℓ≠p into blocks. In level zero, we associate to each block a replacement for the Iwahori–Hecke algebra which provides a Morita equivalence as in the complex case. Additionally, we explain how this gives rise to a description of an arbitrary Gn-block in terms of simple Gm-blocks (for m⩽n), parallelling the approach of Bushnell and Kutzko in the complex setting.