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Geometric structure in smooth dual and local Langlands conjecture. (English) Zbl 1371.11097
Summary: 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.

##### MSC:
 11F85 $$p$$-adic theory, local fields 22E50 Representations of Lie and linear algebraic groups over local fields 11R39 Langlands-Weil conjectures, nonabelian class field theory 20G05 Representation theory for linear algebraic groups
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