A representation of the chamber set Hecke algebra of an affine building
type: Item
instance of: scholarly article
title: A representation of the chamber set Hecke algebra of an affine building
zbMATH Open document ID: 1310.51010
published in: Ricerche di Matematica
publication date: 2014-12-08T00:00:00Z
review text: Helgason's conjecture that any joint-eigenfunction of all invariant differential operators on a symmetric space can be given by the Poisson integral can be re-formulated for affine buildings in terms of their Hecke algebras. In a series of articles, the authors have dealt with affine buildings of rank 3 and showed, case by case, that Helgason's conjecture fails with respect to one type of special vertices or all types of special vertices if and only if a certain matrix \(\mathbb{P}\) is singular. In the paper under review, the authors introduce a finite-dimensional representation of a particular subalgebra of the Hecke algebra of an affine building. They then use this to show that the non-singularity of the matrix \(\mathbb{P}\) is equivalent to a condition for the bijectivity of the Poisson integral found by \textit{S. Kato} [Math. Ann. 257, 1--7 (1981; Zbl 0452.43014)] in the case of buildings from \(p\)-adic reductive groups.
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 6378227
zbMATH Keywords: affine building
zbMATH Keywords: Poisson transform
zbMATH Keywords: Hecke algebra
zbMATH Keywords: representation
author: Anna Maria Mantero
author: Anna Zappa
MaRDI profile type: Publication
full work available at URL: s11587-013-0155-7
OpenAlex ID: W2012675994
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DOI: 10.1007/S11587-013-0155-7
description: scientific article; zbMATH DE number 6378227