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

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 ${\mathrm{\Sigma }}_0$, the Knop root system ${\tilde{\mathrm{\Sigma }}}_0$, and the Macdonald root system ${\mathrm{\Sigma }}_1$, in terms of Galois actions on the absolute roots $\mathrm{\Phi }$; 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.