  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 $F$.
