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 $Sigma_0$, 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.