Pro-p Iwahori Hecke algebras and the dual Vinberg monoid

TL;DR

This paper explores the connection between pro-p Iwahori Hecke algebras of split reductive groups over p-adic fields and the Vinberg monoid of the dual group, leading to new parametrizations of the algebra's center.

Contribution

It establishes a novel relationship between pro-p Iwahori Hecke algebras and the Vinberg monoid, and generalizes Galois representation parametrizations for higher dimensions.

Findings

Relation between H and Vinberg monoid of the dual group

Parametrization of Spec Z by semisimple Galois representations for GL(n)

Generalization of the n=2 case to higher dimensions

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.