Structure theory: extended groups and Whittaker models

Abstract

Part of the goal of the Langlands classification is a parametrization of the representations of real forms of G in terms of L-groups. A difficulty with this goal is that several different pairs (real form, representation) may be isomorphic. The basic example is (SL(2, ℝ),discrete series), where we may have a holomorphic or an antiholomorphic discrete series representation with the same infinitesimal character. Even though the notion of strong real form allows us to separate these pairs (Lemma 1.15), it still gives no reason to prefer one over another. The L-group parameters we find for these representations (typically local systems of some kind) do include a distinguished parameter (a trivial local system). In order to establish a parametrization like Theorem 1.18, we therefore need (roughly speaking) a way to specify a preferred representation in each L-packet.