Dual Groups and Langlands Functoriality

Langlands never separated the Langlands conjectures for GL n from his general principle of functoriality [30]. In particular, he formulated a correspondence between certain Galois representations and admissible or automorphic representations for any connected reductive algebraic group G. For GL n there was a correspondence between certain n-dimensional Galois representations, that is, representations into GL n (ℂ), and admissible representations of GL n (k) or automorphic representations of GL n (A) [4]. For general G we understand what to replace the automorphic side with: admissible representations of G(k) or automorphic rep-resentations of G(A).