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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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