Langlands’ classification of representations of real tori

SupposeG to be a quasi-split reductive group defined over F . To it is associated a connected complex group G∨ whose root datum is dual to that ofG, and the structure ofG determines an algebraic action ofG(F/F ) on G∨. Define G to be the semidirect productG∨⋊WF . Langlands has conjectured a partition of a large part of the set of irreducible representations ofG(F ) into finite packets (calledL packets, because each constituent in a packet is associated to the same local L-function). The conjecture is that these are parametrized by certain continuous homomorphisms from WF to G compatible with the projection from G to WF . The bijection should relate in a natural way to the local factors of L-functions Langlands associates to automorphic forms. If F isC orR this partitions the entire set of irreducible representations, but if F is p-adic one must introduce a group slightly larger thanWF in order to account for certain l-adic representations of the Galois group.