The Langlands classification without L-groups

Abstract

In this chapter we recall the “elementary” version of the Langlands classification of representations, in which L-groups do not appear (Theorem 11.14 below). Because some of the groups we consider (such as G(ℝ)^can) are not precisely groups of real points of connected algebraic groups, we need to formulate this result in a slightly more general setting. With possible generalizations in mind, we allow even nonlinear groups. The class of groups we consider is essentially the one in section 5 of [50]. (The only difference is that Springer allows G to be disconnected, and imposes an additional technical hypothesis that is empty if G is connected.) We refer the reader to [50] for basic structural facts and further references.