Japan. J. Math. 9, 99--136 (2014)

Geometric structure in smooth dual and local Langlands conjecture

A.-M. Aubert, P. Baum, R. Plymen and M. Solleveld

Abstract: This expository paper first reviews some basic facts about $p$-adic fields, reductive $p$-adic groups, and the local Langlands conjecture. If $G$ is a reductive $p$-adic group, then the smooth dual of $G$ is the set of equivalence classes of smooth irreducible representations of $G$. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert--Baum--Plymen--Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.