Irreducible characters of the metaplectic group II: Functoriality

We extend our previous work [RT] and establish a complete duality theory for the (nonalgebraic) real metaplectic group. As a consequence, we obtain an intrinsic local Langlands conjecture for this group and, in particular, develop a geometric theory of endoscopic lifting. We also investigate the behavior of this formalism with theta lifting to equal-size orthogonal groups, and prove that for the kinds of infinitesimal character for which stability is empty, theta lifting preserves Kazhdan-Lusztig character formulas. Finally we interpret a character lifting due to Adams as an instance of functoriality for Mp(2n, ℝ).