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

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 échelonnage root system ${\Sigma }_0$, the Knop root system ${\tilde{\Sigma }}_0$, 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.

28 pages. Final version; to appear in Representation Theory