Pro-p Iwahori Hecke algebras and the dual Vinberg monoid
Tobias Schmidt
TL;DR
The paper develops a bridge between mod-$p$ representation theory of $p$-adic groups and dual groups via the Vinberg monoid, revealing a stratified link between centers of pro-$p$ Iwahori Hecke algebras and mod-$p$ Galois representations valued in the dual group. It constructs and analyzes the dual Vinberg/Zhu monoid, its toral submonoid and Vinberg section, and proves a $q$-Bernstein–Lusztig isomorphism that identifies the Hecke center with functions on a torus-augmented Vinberg semigroup, ultimately producing a Galois-parametrized description of central Hecke strata for GL$_n$ in the unramified case. The work extends known GL$_2$ results to GL$_n$, providing a stratified correspondence indexed by Levi subgroups and connecting inertial types with dual-torus data via Jantzen parametrization and Deligne–Lusztig theory. This framework offers a path toward a modular Langlands-type parametrization of mod-$p$ Hecke modules through dual-group Galois data, with potential applications to the categorical mod-$p$ Langlands program and explicit GL$_n$ cases.
Abstract
Let G be a split reductive group over the integers, F a p-adic local field with residue field Fq. We relate the pro-p-Iwahori Hecke algebra H of G(F) over Fq to the Vinberg monoid of the dual group and study this relation. As an application, in the GL(n)-case and for F/Qp unramified, we derive a parametrization of SpecZ by semisimple n-dimensional representations of the absolute Galois group of F, generalizing the known case n = 2. Here Z denotes the center of H.