Un cas simple de correspondance de Jaquet-Langlands modulo l

Jean-Francois Dat (1)

1 IMJ - Institut de Mathématiques de Jussieu

Let G be a general linear group over a p-adic field and let D^* be an anisotropic inner form of G. The Jacquet-Langlands correspondence between irreducible complex representations of D^* and discrete series of G does not behave well with respect to reduction modulo a prime l different from p. However we show that the Langlands-Jacquet transfer, from the Grothendieck group of admissible l-adic representations of G to that of D^* is compatible with congruences and reduces modulo l to a similar transfer for l-modular representations, which moreover can be characterized by some Brauer characters identities. Studying more carefully this transfer, we deduce a bijection between irreducible l-modular representations of D^* and ''super-Speh'' l-modular representations of G. Via reduction mod l, this latter bijection is compatible with the classical Jacquet-Langlands correspondence composed with the Zelevinsky involution. Finally we discuss the question whether our Langlands-Jacquet transfer sends irreducibles to effective virtual representations up to a sign. This is related to a possible cohomological realization of this transfer, and presumably boils down to some unknown properties of parabolic affine Kazhdan-Lusztig polynomials.

Mots clés: Jacquet-Langlands correspondence; modular representations : p-adic groups

Domaines: Théorie des représentations [math.RT]; Théorie des nombres [math.NT]

Jean-Francois Dat. Un cas simple de correspondance de Jaquet-Langlands modulo l. 2010. ⟨hal-00471865⟩