Un cas simple de correspondance de Jacquet‐Langlands modulo ℓ

By Justice Dat, M. Vignéras. 2010

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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≠p. However, we show that the Langlands–Jacquet transfer, from the Grothendieck group of admissible Q¯ℓ ‐representations of G to that of D× is compatible with congruences and reduces modulo ℓ to a similar transfer for F¯ℓ ‐representations, which moreover can be characterized by some Brauer characters identities. Studying this transfer more carefully, we deduce a bijection between irreducible F¯ℓ ‐representations of D× and ‘super‐Speh’ F¯ℓ ‐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 Vignéras' first construction of Brauer characters for p‐adic groups. It follows Harish‐Chandra's classical approach while our construction uses resolutions on the building.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd0146ebc3589d504ae54ca68dc229187a7b86e9