Curtis homomorphisms and the integral Bernstein center for GLn

Helm, David

doi:10.2140/ant.2020.14.2607

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

David Helm

Vol. 14 (2020), No. 10, 2607–2645

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 ${GL}_{n} (F)$ (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 $m \leq n$ ), together with the strong version of the conjecture for $m < n$ , implies the strong conjecture for ${GL}_{n}$ . 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).

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

Vol. 14, No. 10, 2020