Strongly harmonic forms for representations in the discrete series
type: Item
instance of: scholarly article
title: Strongly harmonic forms for representations in the discrete series
zbMATH Open document ID: 0914.22017
published in: Journal of Functional Analysis
publication date: 1999-06-13T00:00:00Z
review text: Let \(G\) be a semisimple connected Lie group with finite centre, \(K\) a maximal compact subgroup with the same rank as \(G\) and let \(T \subset K\) be a Cartan subgroup of \(G\). In this case \(G\) has a nonempty discrete series, having representations which are associated to regular elliptic coadjoint orbits \(\mathcal O = G/T\). The quotient \(G/T\) is endowed with two \(G\)-invariant hermitian forms, one indefinite, the other positive definite. \textit{W. Schmid} used the latter metric in [Representation theory and harmonic analysis on semisimple Lie groups, Math. Surv. Monogr. 31, 223--286 (1989; Zbl 0744.22016) and Ann. Math. (2) 103, 375--394 (1976; Zbl 0333.22009)] to construct discrete series representations in the space of \(L^{2}\) harmonic forms. In this paper the author considers the formal adjoint operator \(\overline \partial^{*}_{\text{inv}}\) to the standard Dolbeault operator \(\overline {\partial}\) with respect to the invariant indefinite hermitian form on \(G/T\) and she proves that every \(K\)-finite Dolbeault cohomology class admits a strongly harmonic representative, i.e. a representative in \(\ker \overline {\partial} \cap \ker \overline {\partial}^{*}_{\text{inv}}\). This provides explicit integral formulas for the harmonic representatives.
reviewed by: Sergio Console
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 1271584
zbMATH Keywords: harmonic forms
zbMATH Keywords: discrete series
zbMATH Keywords: coadjoint orbits
zbMATH Keywords: semisimple connected Lie group
author: L. Barchini
MaRDI profile type: Publication
full work available at URL: jfan.1998.3327
OpenAlex ID: W2086283480
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DOI: 10.1006/JFAN.1998.3327
description: scientific article; zbMATH DE number 1271584