Curtis homomorphisms and the integral Bernstein center for GL_n

TL;DR

This paper proposes two Galois-theoretic conjectures describing the integral Bernstein center for GL_n over p-adic fields, establishing an inductive approach that links weaker and stronger versions and connects to the local Langlands correspondence.

Contribution

It introduces two conjectures that characterize the integral Bernstein center for GL_n in Galois-theoretic terms and proves their equivalence through an inductive argument.

Findings

Weak conjecture for m ≤ n implies strong conjecture for n

Strong conjecture for n-1 implies weak conjecture for n

Description of the Bernstein center in Galois terms supports local Langlands in families

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.