On modular representations of inner forms of $\mathrm{GL}_n$ over a local non-archimedean field

Let $\mathrm{F}$ be a local non-archimedean field of residue characteristic $p$ and $\overline{\mathbb{F}}_\ell$ an algebraic closure of a finite field of characteristic $\ell \neq p$. We extend the results of Lapid and M\'inguez concerning $\square$-irreducible representations of inner forms of $\mathrm{GL}_n(\mathrm{F})$ to representations over $\overline{\mathbb{F}}_\ell$. As applications, we compute the Godement-Jacquet $L$-factor for any smooth irreducible representation over $\overline{\mathbb{F}}_\ell$ and show that the local factors of a representation agree with the ones of its $\mathrm{C}$-parameter defined by Kurinczuk and Matringe. Moreover, we reprove that the classification of irreducible representations via multisegments due to Vign\'eras and M\'inguez-S\'echerre is indeed exhaustive without using the classification of Ariki and Mathas of simple modules of Hecke algebras. Finally, we characterize the irreducible constituents of certain parabolically induced representations, as was already done by Zelevinsky over $\mathbb{C}$.