Geometric structure in smooth dual and local Langlands conjecture

Anne Marie Aubert, Paul Baum, Roger Plymen, Maarten Solleveld

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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.

Original languageEnglish (US)
Pages (from-to)99-136
Number of pages38
JournalJapanese Journal of Mathematics
Issue number2
Publication statusPublished - Sep 1 2014


All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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