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 non-archimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the { μ } \{ \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 Σ 0 \Sigma _0 , the Knop root system Σ ~ 0 \widetilde {\Sigma }_0 and the Macdonald root system Σ 1 \Sigma _1 , in terms of Galois actions on the absolute roots Φ \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. The latter gives an explicit form of the test function conjecture for general Shimura varieties with parahoric level structure.