The Spherical Dual for p-adic Groups

Abstract

The local Langlands conjectures have played a very significant role in the study of the representation theory of reductive algebraic groups. Roughly they say that the parametrization of equivalence classes of irreducible representations should be given in terms of conjugacy classes of continuous homomorphisms of the Weil group W F into the dual group L G. When G is split, the local Langlands conjectures say basically that the parametrization of equivalence classes of irreducible representations should be given in terms of conjugacy classes of homomorphisms of the Weil group W F into the dual group L G. More precisely, let V G be the connected complex group with root data dual to the root data of G. Then consider V G conjugacy classes of continuous homomorphisms $$\phi :W_\mathbb{R} \to ^L G$$ (0.1) , such that the image consists of semisimple elements. In the real case, these conjectures were crucial for the classification of admissible irreducible (g, K) modules in the work of Langlands, Shelstad, Knapp—Zuckerman and Vogan. In the p-adic case they play a significant role in the work of Kazhdan—Lusztig and Lusztig. There is a technical modification in that one considers maps of the Weil—Deligne—Langlands group, $$\phi :W_\mathbb{F} \times SL(2) \to \,^L G$$ (0.2) .