On modular representations of inner forms of \(\textrm{GL}_n\) over a local non-archimedean field

Let \(\textrm{F}\) $\text{F}$ be a local non-archimedean field of residue characteristic p and \(\overline{\mathbb {F}}_\ell \) ${\overline{\mathbb{F}}}_{\ell }$ an algebraic closure of a finite field of characteristic \(\ell \ne p\) $\ell \ne p$ . We extend the results of [22] concerning \(\square \) $\square$ -irreducible representations of inner forms of \(\textrm{GL}_n(\textrm{F})\) ${\text{GL}}_n(\text{F})$ to representations over \({\overline{\mathbb {F}}_\ell }\) ${\overline{\mathbb{F}}}_{\ell }$ . As applications, we compute the Godement-Jacquet L-factor for any smooth irreducible representation over \({\overline{\mathbb {F}}_\ell }\) ${\overline{\mathbb{F}}}_{\ell }$ and show that the local factors of a representation agree with the ones of its \(\textrm{C}\) $\text{C}$ -parameter defined in [19]. Moreover, we reprove that the classification of irreducible representations via multisegments due to Vignéras and Mínguez-Sécherre is indeed exhaustive without using the results of [2]. Finally, we characterize the irreducible constituents of certain parabolically induced representations, as was already done by Zelevinsky over \(\mathbb {C}\) $\mathbb{C}$ .