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2020 Curtis homomorphisms and the integral Bernstein center for $\mathrm{GL}_n$
David Helm
Algebra Number Theory 14(10): 2607-2645 (2020). DOI: 10.2140/ant.2020.14.2607

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 mn), 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 n1 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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David Helm. "Curtis homomorphisms and the integral Bernstein center for $\mathrm{GL}_n$." Algebra Number Theory 14 (10) 2607 - 2645, 2020. https://doi.org/10.2140/ant.2020.14.2607

Information

Received: 13 November 2018; Revised: 11 March 2020; Accepted: 30 June 2020; Published: 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4190413
Digital Object Identifier: 10.2140/ant.2020.14.2607

Subjects:
Primary: 11F33
Secondary: 11F70 , 22E50

Keywords: $p$-adic groups , Langlands correspondence , modular representation theory

Rights: Copyright © 2020 Mathematical Sciences Publishers

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