Abstract
We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for (that is, the center of the category of smooth -modules, for a -adic field and an algebraically closed field of characteristic different from ) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for ), together with the strong version of the conjecture for , implies the strong conjecture for . In a companion paper (Invent. Math. 214:2 (2018), 999–1022) we show that the strong conjecture for implies the weak conjecture for ; 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. 47:4 (2014), 655–722).
Citation
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