Un cas simple de correspondance de Jacquet-Langlands modulo l

TL;DR

This paper studies the compatibility of the Langlands-Jacquet transfer with modulo l reduction for representations of p-adic groups, establishing a bijection between certain modular representations and exploring its properties.

Contribution

It demonstrates that the Langlands-Jacquet transfer is compatible with modulo l reduction and characterizes the resulting transfer using Brauer characters, connecting modular representations.

Findings

The transfer reduces modulo l to a similar transfer for l-modular representations.

A bijection is established between irreducible l-modular D^* representations and super-Speh representations of G.

The transfer's behavior with respect to virtual representations and signs is discussed.

Abstract

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 l\neq 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. Our arguments rely on Vigneras' classification of representations of G. 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.