#### 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 ${GL}_{n}\left(F\right)$ (that is, the center of the category of smooth $W\left(k\right)\left[{GL}_{n}\left(F\right)\right]$-modules, for $F$ a $p$-adic field and $k$ an algebraically closed field of characteristic $\ell$ different from $p$) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for $m\le n$), together with the strong version of the conjecture for $m, 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. $\left(4\right)$ 47:4 (2014), 655–722).

##### Keywords
Langlands correspondence, modular representation theory, $p$-adic groups
##### Mathematical Subject Classification 2010
Primary: 11F33
Secondary: 11F70, 22E50