Functional Hecke algebras and simple Bernstein blocks of a p-adic GL_n in non-defining characteristic

Let $G_n={\mathrm{GL}}_n(F)$, where $F$ is a non-archimedean local field with residue characteristic $p$ and where $n=2k$ is even. In this article, we investigate a question occurring in the decomposition of the category of $\mathrm{\ell }$-modular smooth representations of $G_n$ into Bernstein blocks (where $\mathrm{\ell }\mathrm{\ne }p$). The easiest block not investigated in \cite{guiraud} is the one defined by the standard parabolic subgroup with Levi factor $M={\text{GL}}_k(F)\times {\text{GL}}_k(F)$ and by an $M$-representation of the form ${\pi }_0⊠{\pi }_0$ with ${\pi }_0$ a supercuspidal ${\text{GL}}_k(F)$-representation. This block is Morita equivalent to a Hecke algebra which we can describe as a twisted tensor product of a finite Hecke algebra (i. e. a Hecke algebra occurring in the representation theory of the finite group ${\text{GL}}_k(p^{\alpha })$ in non-defining characteristic $\mathrm{\ell }$) and the group ring of ${\mathbb{Z}}^2$. This enables us to describe how a conjectured connection between finite Hecke algebras (which is similar to a connection postulated by Brou'e in \cite{Broue}) would lead to an equivalence between the described block and the unipotent block of ${\mathrm{GL}}_2(F^k)$, where $F^k$ is the unramified extension of degree $k$ over $F$.