What question did this study set out to answer?

The study aims to extend findings on irreducible representations of inner forms of GL_n over local non-archimedean fields and their modular aspects.

June 15, 2026Open Access

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

Key Points

The study aims to extend findings on irreducible representations of inner forms of GL_n over local non-archimedean fields and their modular aspects.
Extending previous results on irreducible representations to the context of algebraic closures of finite fields.
Computing the Godement-Jacquet L-factor for smooth irreducible representations over algebraic closures.
Revising the classification of irreducible representations using multisegments without prior results.
Confirmed that local factors of representations align with those of their C-parameters as defined in earlier studies.
Demonstrated exhaustiveness of multisegment classification for irreducible representations.
Characterized irreducible constituents of certain parabolically induced representations.

Abstract

Abstract Let F F be a local non-archimedean field of residue characteristic p and F_ 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 {F_ } F ¯ ℓ. As applications, we compute the Godement-Jacquet L -factor for any smooth irreducible representation over {F_ } 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.

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