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

Abstract Let $$\textrm{F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>F</mml:mtext> </mml:math> be a local non-archimedean field of residue characteristic p and $$\overline{\mathbb {F}}_\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>F</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> an algebraic closure of a finite field of characteristic $$\ell \ne p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> . We extend the results of [22] concerning $$\square $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>□</mml:mo> </mml:math> -irreducible representations of inner forms of $$\textrm{GL}_n(\textrm{F})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>F</mml:mtext> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> to representations over $${\overline{\mathbb {F}}_\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>F</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> . As applications, we compute the Godement-Jacquet L -factor for any smooth irreducible representation over $${\overline{\mathbb {F}}_\ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>F</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:math> and show that the local factors of a representation agree with the ones of its $$\textrm{C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>C</mml:mtext> </mml:math> -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 $$\mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> .