On modular representations of inner forms of GLₙ over a local non-archimedean field

Let F F be a local non-archimedean field of residue characteristic p and F̄_ F ¯ ℓ an algebraic closure of a finite field of characteristic ≠ p ℓ ≠ p . We extend the results of [22] concerning □ -irreducible representations of inner forms of GLₙ(F) GL n ( F ) to representations over F̄_ F ¯ ℓ . As applications, we compute the Godement-Jacquet L -factor for any smooth irreducible representation over F̄_ F ¯ ℓ and show that the local factors of a representation agree with the ones of its C 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 C C .