Curtis homomorphisms and the integral Bernstein center for GL_n

Abstract

We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GL_n(F) (that is, the center of the category of smooth W(k)[GL_n(F)]-modules, for F a p-adic field and k an algebraically closed field of characteristic l different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for m at most n) implies the strong version of the conjecture. In a companion paper [HM] we show that the strong conjecture for n-1 implies the weak conjecture for n; thus the two papers together give an inductive proof of both conjectures. The upshot is a description of the integral Bernstein center for GL_n in purely Galois-theoretic terms; previous work of the author shows that such a description implies the conjectural "local Langlands correspondence in families" of Emerton and the author.

Final accepted version. 36 pages. Note that the published version of Helm-Moss, "Converse theorems and the local Langlands correspondence in families" references an earlier version of this paper, and the section numbering has changed; in particular sections 9,10, and 11 of the referenced version correspond to sections 8,9, and 10 of the current version, respectively

References in corpus (1)

• The Breuil--Mézard conjecture when $l\ne p$

Cited by in corpus (5)

• Coherent Springer theory and the categorical Deligne-Langlands correspondence
• The Jacobson--Morozov morphism for Langlands parameters in the relative setting
• On endomorphism algebras of Gelfand-Graev representations II
• Generic local deformation rings when $l\ne p$
• Irreducible components of the moduli space of Langlands parameters