Dualities for root systems with automorphisms and applications to non-split groups

TL;DR

This paper develops dualities for root systems with automorphisms and applies them to non-split reductive groups over local fields, proving conjectures and describing root systems and centers in a unified framework.

Contribution

It introduces elementary dualities for root systems with automorphisms and applies these to characterize admissible sets, root systems, and centers of Hecke algebras for non-split groups.

Findings

Proved a conjecture characterizing extremal elements of admissible sets.

Provided a uniform description of various root systems via Galois actions.

Constructed the geometric basis of the center of parahoric Hecke algebras.

Abstract

This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over nonarchimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the $\{ \mu \}$-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Tits \'{e}chelonnage root system $\Sigma_0$, the Knop root system $\widetilde{\Sigma}_0$, and the Macdonald root system $\Sigma_1$, in terms of Galois actions on the absolute roots $\Phi$; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis.