Langlands classification for L-parameters

Abstract

Let F be a non-archimedean local field and G=G(F) the group of F-rational points of a connected reductive F-group. The Langlands classification of complex irreducible admissible representations of G gives a parametrization of the set of all such representations in terms of triples (P,σ,ν), where P⊂G is a standard F-parabolic subgroup, σ is an irreducible tempered representation of the standard Levi-group MP and ν∈R⊗X⁎(MP) is regular and positive with respect to P. In this paper we consider Langlands' L-parameters [ϕ] which conjecturally will serve as a system of parameters for the representations π and which are equivalence classes of L-homomorphisms ϕ of the absolute Galois group Γ=Gal(F‾|F) with image in Langlands' L-group GL. Our goal is to establish a classification of L-parameters [ϕ] in terms of triples (P,[ϕb],ν) such that (P,ν) is as before and [ϕb] is a bounded L-parameter of MP.In an Appendix we also deal with the archimedean case F=R,C. This case is easier because the Weil group WR has a natural polar decomposition WR=R+××WR0.