The action of intertwining operators on spherical vectors in the minimal principal series of a reductive symmetric space
type: Item
instance of: scholarly article
title: The action of intertwining operators on spherical vectors in the minimal principal series of a reductive symmetric space
zbMATH Open document ID: 0914.22018
DOI: 10.1016/S0019-3577(97)81814-9
published in: Indagationes Mathematicae. New Series
publication date: 1999-05-05T00:00:00Z
review text: Let \(G\) be a real semisimple Lie group with finite center, \(\sigma\) an involution of \(G\) and \(H\) an open subgroup of the group \(G^\sigma\) of fixed points for \(\sigma\). Moreover, let \(\Theta\) be a Cartan involution commuting with \(\sigma\). The minimal principal series which appear in the most continuous part of the Plancherel decomposition of \(L^2(G/H)\) are parabolically induced representations \(\pi_{\xi,\lambda}=\text{ind}^G_P(\xi\oplus\lambda\oplus 1)\), where \(P=MAN\) is a minimal \(\sigma\Theta\)-stable parabolic subgroup of \(G,\xi\) a finite dimensional unitary representation of \(M\) and \(\lambda\in{\mathfrak a}^*_{q\mathbb{C}}\) the space of complex characters of \(A\) with \(\sigma\nu=-\nu\). Let \(C^{-\infty}(P:\xi:\lambda)\) denote the space of generalized sections of the homogeneous vector bundle in which \(\pi_{\xi,\lambda}\) is naturally realized and let \(C^{-\infty}(P:\xi:\lambda)^H\) be its \(H\)-fixed elements. In the theory an important role is played by the standard intertwining operator \(A(Q:P:\xi:\lambda):C^{-\infty}(P:\xi:\lambda)\to C^{-\infty}(Q:\xi:\lambda)\), with \(Q\) a parabolic subgroup associated to \(P\). By equivariance the standard intertwining operator maps the spherical vectors for \(\pi_{P,\xi,\lambda}\) to those for \(\pi_{Q,\xi,\lambda}\). In [Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, 359-412 (1988; Zbl 0714.22009)] the author had established the existence of a finite-dimensional Hilbert space \(V(\xi)\) and a linear map \(j(P:\xi:\lambda)\to C^{-\infty}(P:\xi:\lambda)^H\) depending meromorphically on \(\lambda\in{\mathfrak a}^*_{q\mathbb{C}}\) and bijective for generic \(\lambda\). He also established the existence of a unique endomorphism \(B(Q:P:\xi:\lambda)\) of \(V(\xi)\) such that \[ A(Q:P:\xi:\lambda)\circ j(P:\xi:\lambda)=j(P:\xi:\lambda)\circ B(Q:P:\xi:\lambda). \] In the paper mentioned above the author proved that \(B(Q:P:\xi:-\overline\lambda)^*=B(Q:P:\xi:\lambda)\). For part of the argument lea…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 1136775
zbMATH Keywords: real semisimple Lie group
zbMATH Keywords: Plancherel decomposition
zbMATH Keywords: unitary representation
zbMATH Keywords: intertwining operator
author: Erik P. van den Ban
reviewed by: Jean Ludwig
MaRDI profile type: Publication
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full work available at URL: s0019-3577(97)81814-9
OpenAlex ID: W2138233975
description: scientific article; zbMATH DE number 1136775