\(H\)-fixed distribution vectors for generalized principal series of reductive symmetric spaces and meromorphic continuation of Eisenstein integrals
type: Item
instance of: scholarly article
title: \(H\)-fixed distribution vectors for generalized principal series of reductive symmetric spaces and meromorphic continuation of Eisenstein integrals
zbMATH Open document ID: 0785.22014
DOI: 10.1007/BF01232043
author: Jean-Luc Brylinski
author: Patrick Delorme
published in: Inventiones Mathematicae
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/144037
review text: Let \(G\) be the group of real points of a Zariski-connected linear algebraic group defined over \(\mathbb{R}\), equipped with an involution \(\sigma\) defined over \(\mathbb{R}\). Let \(H\) be the group of fixed points of \(\sigma\) in \(G\). Then \(G/H\) is a reductive symmetric space. Let \(\theta\) be a Cartan involution of \(G\) commuting with \(\sigma\), and let \(P\subset G\) be a parabolic subgroup of \(G\) which is \(\sigma\circ\theta\)-stable. Then \(P\) has a \(\sigma\)-Langlands decomposition \(P=MAN\). Here \(N\) is the unipotent radical of \(P\), and \(L=P\cap \theta P\) the \(\theta\)-stable Levi component. Moreover, \(A\) is the subgroup of the usual Langlands \(A\) consisting of elements \(a\) with \(\sigma a=a^{-1}\). Suppose that the reductive symmetric space \(M/M\cap H\) satisfies Flensted-Jensen's rank condition so that it possesses a discrete series representation \(\delta\). If \(\nu\in{\mathfrak a}_{\mathbb{C}}^*\), then the representation \(\pi_{\delta,\nu}\) of \(G\) induced from the representation \(\delta\otimes (\nu+\rho)\otimes 1\) of \(P\) is called a generalized principal series representation of \(G/H\). The authors establish the existence of an \(H\)-fixed distribution vector in \(\pi_{\delta,\nu}\) which depends meromorphically on the parameter \(\nu\). The main interest in such a family stems from its expected contribution to the Plancherel decomposition of \(L^ 2(G/H)\). As a consequence of the above mentioned result the authors establish meromorphic continuation of Eisenstein integrals, i.e. matrix coefficients of \(K\)-finite vectors with \(H\)-fixed distribution vectors in \(\pi_{\delta,\nu}\). The authors also give an application of their result to standard intertwining operators: their distribution kernels are examples of the \(H\)-fixed distribution vectors under consideration, in the setting of the group \(G\) viewed as the symmetric space \(G\times G/\text{diagonal }G\). The present work generalizes earlier work by \text…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 220753
zbMATH Keywords: stable parabolic subgroups
zbMATH Keywords: reductive symmetric space
zbMATH Keywords: generalized principal series representation
zbMATH Keywords: \(H\)-fixed distribution vector
zbMATH Keywords: Plancherel decomposition
zbMATH Keywords: meromorphic continuation of Eisenstein integrals
zbMATH Keywords: matrix coefficients
zbMATH Keywords: intertwining operators
MaRDI profile type: Publication
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OpenAlex ID: W2078812466
description: scientific article; zbMATH DE number 220753