On semisimple ℓ-modular Bernstein-blocks of a p-adic general linear group

Let Gn=GLn(F)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 GnGn over an algebraically closed field of positive characteristic ℓ≠pℓ≠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 GnGn-block in terms of simple GmGm-blocks (for m⩽nm⩽n), parallelling the approach of Bushnell and Kutzko in the complex setting.