Vol. 14, No. 10, 2020

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Curtis homomorphisms and the integral Bernstein center for $\mathrm{GL}_n$

David Helm

Vol. 14 (2020), No. 10, 2607–2645
Abstract

We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GLn(F) (that is, the center of the category of smooth W(k)[GLn(F)]-modules, for F a p-adic field and k an algebraically closed field of characteristic different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for m n), together with the strong version of the conjecture for m < n, implies the strong conjecture for GLn. In a companion paper  (Invent. Math. 214:2 (2018), 999–1022) 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 Bernstein center in purely Galois theoretic terms; previous work of the author shows that this description implies the conjectural “local Langlands correspondence in families” of  (Ann. Sci. Éc. Norm. Supér. (4) 47:4 (2014), 655–722).

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Keywords
Langlands correspondence, modular representation theory, $p$-adic groups
Mathematical Subject Classification 2010
Primary: 11F33
Secondary: 11F70, 22E50
Milestones
Received: 13 November 2018
Revised: 11 March 2020
Accepted: 30 June 2020
Published: 19 November 2020
Authors
David Helm
Department of Mathematics
Imperial College London
United Kingdom