Un cas simple de correspondance de Jacquet-Langlands modulo l

Let G be a general linear group over a p-adic field and let D-x be an anisotropic inner form of G. The Jacquet-Langlands correspondence between irreducible complex representations of D-x and discrete series of G does not behave well with respect to reduction modulo l not equal p. However, we show that the Langlands-Jacquet transfer, from the Grothendieck group of admissible (Q) over bar (e)-representations of G to that of D-x is compatible with congruences and reduces modulo l to a similar transfer for (F) over bar (e)-representations, which moreover can be characterized by some Brauer characters identities. Studying this transfer more carefully, we deduce a bijection between irreducible (F) over bar (e)-representations of D-x and 'super-Speh' (F) over bar (e)-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 presumably boils down to some unknown properties of parabolic affine Kazhdan-Lusztig polynomials. In the appendix, we reproduce Vigneras' first construction of Brauer characters for p-adic groups. It follows Harish-Chandra's classical approach while our construction uses resolutions on the building.