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

Let $G_n={\mathrm{GL}}_n(F)$, where $F$ is a non-archimedean local field with residue characteristic $p$. Our starting point is the Bernstein-decomposition of the representation category of $G_n$ over an algebraically closed field of characteristic $\mathrm{\ell }\mathrm{\ne }p$ into blocks. In level zero, we associate to each block a replacement for the Iwahori-Hecke algebra which provides a Morita-equivalence just as in the complex case. Additionally, we will explain how this gives rise to a description of an arbitrary $G_n$-block in terms of simple $G_m$-blocks (for $m\le n$), paralleling the approach of Bushnell and Kutzko in the complex setting.