Complex structure on the smooth dual of \(GL(n)\)
type: Item
instance of: scholarly article
title: Complex structure on the smooth dual of \(GL(n)\)
zbMATH Open document ID: 1012.22031
author: Jacek Brodzki
author: Roger Plymen
published in: Documenta Mathematica
publication date: 2002-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/122647
full work available at URL: 04.html
review text: Let \(G= GL(n)\) be the general linear group over a \(p\)-adic field. The set of equivalence classes of irreducible smooth representations of \(G\) is called the smooth dual of \(G\) and denoted by \(\Pi(G)\). The local Langlands correspondence for \(GL(n)\) is a bijection between \(\Pi(G)\) and the set of Langlands parameters \(\Phi(G)\). The authors introduced in [\textit{J. Brodzki} and \textit{R. Plymen}, C. R. Acad. Sci. Paris, Sér. A 331, 213-218 (2000; Zbl 0970.46002)] a complex structure on the set of Langlands parameters of \(G\). This gives, via the local Langlands correspondence, a complex structure on the smooth dual of \(G\). In the paper under review, the authors describe in detail the complex structure on \(\Phi(G)\). Let \(\Pi(\Omega)\) denote a Bernstein component of \(\Pi(G)\) and let \(\mathcal{H}(\Omega)\) denote a Bernstein ideal in the Hecke algebra \(\mathcal{H}(G)\). The authors prove that the periodic cyclic homology of \(\mathcal{H}(\Omega)\) is isomorphic to the de Rham cohomology of \(\Pi(\Omega)\). They show how the structure of the variety \(\Pi(\Omega)\) is related to Xi's affirmation of a conjecture of Lusztig for \(GL(n,\mathbb{C})\). The smooth dual \(\Pi(G)\) admits a deformation retraction onto the tempered dual \(\Pi^t(G)\).
reviewed by: Dubravka Ban
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 1748375
zbMATH Keywords: \(p\)-adic \(GL(n)\)
zbMATH Keywords: smooth dual
zbMATH Keywords: Langlands correspondence
zbMATH Keywords: Bernstein variety
zbMATH Keywords: Baum-Connes map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1748375