Level zero types and Hecke algebras for local central simple algebras
type: Item
instance of: scholarly article
title: Level zero types and Hecke algebras for local central simple algebras
zbMATH Open document ID: 1009.22016
author: Martin Grabitz
author: Allan J. Silberger
author: Ernst-Wilhelm Zink
published in: Journal of Number Theory
publication date: 2002-08-25T00:00:00Z
review text: The theory of types for connected reductive \(p\)-adic groups was introduced by C. Bushnell and P. Kutzko. A type is a pair \((K, \tau)\) consisting of a certain open subgroup \(K\) and a certain irreducible representation \(\tau\) of \(K\). For \(G=GL_n(F)\) over the \(p\)-adic field \(F\), a classification of the irreducible smooth representations of \(G\) via types is given in terms of Hecke algebra isomorphisms [\textit{C. Bushnell} and \textit{P. Kutzko}, Ann. Math. Stud. 129 (1993; Zbl 0787.22016)]. Let \(D\) be a central division algebra and \(A^\times=GL_m(D)\) the unit group of a central simple algebra over \(F\). This article concerns the theory of types for \(A^\times\). The authors give a classification of types for all level zero Bernstein components of \(A^\times\) and prove that the Hecke algebras associated to these types are isomorphic to tensor products of Iwahori Hecke algebras, as in the split case \(G=GL_n(F)\) due to Bushnell and Kutzko. Level zero types were also studied by \textit{L. Morris} [Compos. Math. 118, No. 2, 135-157 (1999; Zbl 0937.22011)].
reviewed by: Dubravka Ban
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 1787789
zbMATH Keywords: level zero types
zbMATH Keywords: Hecke algebras
zbMATH Keywords: local central simple algebras
MaRDI profile type: Publication
OpenAlex ID: W2004254466
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DOI: 10.1006/JNTH.2001.2684
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description: scientific article; zbMATH DE number 1787789