Eigenspaces of operator algebras on the fundamental apartment of an affine building
type: Item
instance of: scholarly article
title: Eigenspaces of operator algebras on the fundamental apartment of an affine building
zbMATH Open document ID: 1310.51011
published in: Ricerche di Matematica
publication date: 2014-12-08T00:00:00Z
review text: In [Ric. Mat. 62, No. 2, 341--362 (2013; Zbl 1310.51010)], the authors considered the Hecke algebra of an affine building of rank 3 and obtained a characterization of the bijectivity of the Poisson transform in terms of the cyclicity of a finite set of vectors under a certain representation of the Hecke algebra introduced there. In the paper under review, the authors' aim is to extend their result to affine buildings of arbitrary rank \(\geq 3\). The key ingredient in achieving this goal is knowledge of the dimension of a certain eigenspace of the operator algebra \(\widetilde{\mathcal{H}}_c(\mathbb{A})\) which is the retraction of the Hecke algebra with respect to a chamber onto a fundamental apartment \(\mathbb{A}\). They proceed by first considering the Hecke algebra of \(\mathbb{A}\), then the retraction with respect to a fixed boundary point and finally \(\widetilde{\mathcal{H}}_c(\mathbb{A})\). It is shown that for each character \(\chi\) of \(\mathbb{A}\) the dimension of the eigenspace in question is finite and at least the order of the Weyl group \(\mathbf{W}\). Equality occurs if and only if the stabilizer \(\mathbf{W}_\chi\) of \(\chi\) is a reflection group.
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 6378272
zbMATH Keywords: affine building
zbMATH Keywords: retraction
zbMATH Keywords: Hecke algebra
zbMATH Keywords: operator algebra
zbMATH Keywords: eigenspace
zbMATH Keywords: Poisson kernel
author: Anna Maria Mantero
author: Anna Zappa
MaRDI profile type: Publication
full work available at URL: s11587-014-0181-0
OpenAlex ID: W2004158394
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DOI: 10.1007/S11587-014-0181-0
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description: scientific article; zbMATH DE number 6378272