Decay of matrix coefficients on reductive homogeneous spaces of spherical type
type: Item
instance of: scholarly article
title: Decay of matrix coefficients on reductive homogeneous spaces of spherical type
zbMATH Open document ID: 1303.22006
author: Bernhard Krötz
author: Eitan Sayag
author: Henrik Schlichtkrull
published in: Mathematische Zeitschrift
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.2943
review text: Let \(Z\) be a homogeneous space \(Z = G/H\) of a real reductive Lie group \(G\) with a reductive subgroup \(H.\) The authors define \(Z\) to be of spherical type if there exists a minimal parabolic subgroup \(P\) such that \(PH\) is open in \(G.\) For a continuous linear representation \((\pi, E)\) of \(G,\) on a Banach space \(E,\) and for a vector \(v \in E\) and \(\eta \in E^\ast\) define the matrix coefficient \[ m_{v, \eta}(g) = \eta(\pi(g^{-1}v) \quad g \in G. \] The paper is concerned with estimates for these matrix coefficients. Assume that \(Z\) is of spherical type and \(E = V^\infty\) is the smooth globalization of a Harish-Chandra module \(V.\) Let \(K\) be a maximal compact subgroup, \(P= MAN\) the Langlands decomposition of the minimal parabolic subgroup and \(A^{+}\) the Weyl chamber associated to \(P.\) Then, for every \(v \in V\) there is a constant \(C > 0\) such that \[ |m_{v, \eta}(g)| \leq C~a^{\Lambda_V} (1+\| \log a\|)^{d_V}, \quad a \in A^{+}, \] where \(\Lambda_V \in \mathfrak{a}^\ast \) and \(d_V \in \mathbb N\) are determined by \(V.\) Several examples are also discussed.
reviewed by: E. K. Narayanan
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 6357848
zbMATH Keywords: Lie group
zbMATH Keywords: representations
zbMATH Keywords: matrix coefficients
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: spherical type
MaRDI profile type: Publication
OpenAlex ID: W2136212737
arXiv ID: 1211.2943
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DOI: 10.1007/S00209-014-1313-7
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description: scientific article; zbMATH DE number 6357848