Let $G_{ n}=\operatorname{ GL}_{ n}(F)$, 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 $\ell$-modular smooth representations of $G_n$ into Bernstein blocks (where $\ell\neq p$). The easiest block not investigated in \cite{ guiraud} is the one defined by the standard parabolic subgroup with Levi factor $M=\GL_k(F) \times \GL_k(F)$ and by an $M$-representation of the form $\pi_0 \boxtimes \pi_0$ with $\pi_0$ a supercuspidal $\GL_k(F)$-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 $\GL_k(p^{ \alpha})$ in non-defining characteristic $\ell$) 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 $\operatorname{ GL}_2(F^k)$, where $F^k$ is the unramified extension of degree $k$ over $F$.
Categories: math.NT