Tempered representations of \(p\)-adic groups: special idempotents and topology
type: Item
instance of: scholarly article
title: Tempered representations of \(p\)-adic groups: special idempotents and topology
zbMATH Open document ID: 1365.22005
author: Peter Schneider
author: Ernst-Wilhelm Zink
published in: Selecta Mathematica. New Series
publication date: 2016-12-27T00:00:00Z
review text: In the present paper, P. Schneider et E.-W. Zink carry on their investigation of the category \(\mathcal M^t(G)\) of isomophism classes of tempered representations of a connected reductive \(p\)-adic group \(G\), in the same way as the study of smooth representations of \(G\) due to J. Bernstein, and to \textit{C. J. Bushnell} and \textit{P. C. Kutzko} [Proc. Lond. Math. Soc. (3) 77, No. 3, 582--634 (1998; Zbl 0911.22014)]. The abelian category \(\mathcal M^t(G)\), equivalently viewed as the category of all non-degenerate modules over the algebra \(\mathcal S(G)\) of Schwartz functions on \(G\), decomposes into a direct product of component categories \(\mathcal M^t(\Theta)\) where \(\Theta\) runs over the connected components of the Harish-Chandra spectrum \(\Omega^t(G)\), that are the unitary inertial classes of discrete pairs [\textit{P. Schneider} and \textit{E.-W. Zink}, Geom. Funct. Anal. 17, 2018--2065 (2008; Zbl 1144.22012)]. On the other hand, each idempotent \(e\) of \(\mathcal S(G)\) defines the full subcategory \(\mathcal M^t_{e}(G)\) of all non-degenerate \(\mathcal S(G)\)-modules \(V\) generated by \(eV\). The idempotent \(e\) is special in \(\mathcal S(G)\) if the category \(\mathcal M^t_{e}(G)\) is abelian. At first, the authors prove that for each special idempotent \(e\) of \(\mathcal S(G)\), there exists a finite set \(\theta\) of component categories, called the support of \(e\), such that the category \(\mathcal M^t_{e}(G)\) is the direct product of the component categories \(\mathcal M^t(\Theta)\) where \(\Theta\) runs over \(\theta\). Conversely, they describe any finite direct product of components, in particular any component \(\mathcal M^t(\Theta)\), as a subcategory defined by a special idempotent \(e\) of \(\mathcal S(G)\). The strategy of proofs is the one of Bushnell and Kutzko [loc. cit.] but the arguments are of analytic nature. Then, in order to characterize the special idempotents of \(\mathcal S(G)\) belonging to the Hecke algebra …
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 6668631
zbMATH Keywords: tempered representations
zbMATH Keywords: \(p\)-adic reductive group
zbMATH Keywords: Schwartz algebra
zbMATH Keywords: special idempotent
MaRDI profile type: Publication
full work available at URL: s00029-016-0274-6
OpenAlex ID: W2553252160
cites work: The principal series of p‐adic groups with disconnected center
cites work: A unitarity criterion for p-adic groups
cites work: Q3999894
cites work: LA FORMULE DE PLANCHEREL POUR LES GROUPES p-ADIQUES. DAPRS HARISH-CHANDRA
cites work: Groupes reductifs
cites work: Q3885948
cites work: Q3941610
cites work: The lattice of integer partitions
cites work: Smooth representations of reductive p -ADIC groups: structure theory via types
cites work: THE ALGEBRAIC THEORY OF TEMPERED REPRESENTATIONS OF p-ADIC GROUPS I. PARABOLIC INDUCTION AND RESTRICTION
cites work: Proof of the Deligne-Langlands conjecture for Hecke algebras
cites work: Caracteres a valeurs dans le centre de Bernstein
cites work: Q4932244
cites work: Q4743014
cites work: The algebraic theory of tempered representations of \(p\)-adic groups. II: Projective generators
cites work: K-types for the tempered components of a p-adic general linear group (With an Appendix by P. Schneider, U. Stuhler: The definition of the tempered category)
cites work: The Langlands quotient theorem for p-adic groups
cites work: Linear algebraic groups.
cites work: Q3358957
cites work: Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
DOI: 10.1007/S00029-016-0274-6
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description: scientific article; zbMATH DE number 6668631