Determination of the Harish-Chandra \(C\)-function for \(SU(n, 1)\) and its application to the construction of the composition series
type: Item
instance of: scholarly article
title: Determination of the Harish-Chandra \(C\)-function for \(SU(n, 1)\) and its application to the construction of the composition series
zbMATH Open document ID: 0873.22006
author: Masaaki Eguchi
author: Shin Koizumi
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-02-23T00:00:00Z
review text: Let \(G\) be a reductive Lie group of the Harish-Chandra class, \(G= \text{KAN}\) its Iwasawa decomposition, \(M\) the centralizer in \(K\), \({\mathfrak A}\) the Lie algebra of \(A\), \(\widehat{K}\) (resp. \(\widehat{M})\) the set of equivalence classes of irreducible unitary representations of \(K\) (resp. \(M\)). For \(\tau\in \widehat{K}\), \(\sigma\in \widehat{M}\), \([\tau:\sigma]\) the multiplicity of \(\sigma\in \widehat{M}\) occurring in \(\tau|_M\), \(\widehat{M}(\tau)= \{\sigma\in\widehat{M}\); \([\tau:\sigma]=1\}\). For \(g\in G\), let \(g=\kappa(g) \exp H(g)n(g)\), \((\kappa(g)\in K\), \(H(g)\in{\mathfrak A}\), \(n(g)\in N)\). For \(\tau\in \widehat{K}\) and \(\lambda\in{\mathfrak A}_{\mathbb{C}}^*\) the integral \[ C_\tau(\lambda)= \int_{\overline{N}} \tau(\kappa(\overline{n}))^{-1} e^{- (\lambda+\rho) (H(\overline{n}))} d\overline{n} \] is known as the Harish-Chandra \(C\)-function associated with \(\tau\in \widehat{K}\). When \(G= \text{SU}(n,1)\), the case considered by the authors, \([\tau,\sigma]=0\) or 1 for any \(\tau\in\widehat{K}\) and \(\sigma\in \widehat{M}\). For \(\tau\in \widehat{K}\) write \(\widehat{M}(\tau)= \{\sigma\in \widehat{M}\), \([\tau,\sigma]=1\}\). Let \(\tau\in \widehat{K}\), \(\sigma\in \widehat{M}(\tau)\), and \(V_\tau\), \(H_\sigma\) the corresponding representation spaces. Choose \(P_\sigma(\tau)\in \Hom_M(V_\tau,H_\sigma)\) such that \(P_\sigma(\tau) P_\sigma(\tau)^*= \sigma(1)\). Since \(P_\sigma(\tau) C_\tau(\lambda)\in \Hom_M(V_\tau,H_\sigma)\) and \(\dim \Hom_M(V(\tau), H(\sigma))=1\), there exists a constant \(C_\tau(\sigma:\lambda)\) such that \(P_\sigma(\tau) C_\tau(\lambda)= C_\tau(\sigma:\lambda) P_\sigma(\tau)\). The meromorphic function \(C_\tau(\sigma:\lambda)\) is the Harish-Chandra \(C\)-function associated with \(\tau\) and \(\sigma\). Its importance for harmonic analysis on \(G\) is well-known. It is connected with intertwining operators. It carries information regarding irreducibility and unitarizabilit…
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 952363
zbMATH Keywords: reductive Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: unitary representations
zbMATH Keywords: Harish-Chandra \(C\)-function
zbMATH Keywords: intertwining operators
zbMATH Keywords: classical groups
zbMATH Keywords: composition series
MaRDI profile type: Publication
full work available at URL: pjaa.72.129
OpenAlex ID: W1984244730
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DOI: 10.3792/PJAA.72.129
description: scientific article; zbMATH DE number 952363