Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces
type: Item
instance of: scholarly article
title: Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces
zbMATH Open document ID: 0784.22003
DOI: 10.1016/0022-1236(92)90030-M
author: Frederick P. Greenleaf
author: Lawrence J. Corwin
published in: Journal of Functional Analysis
publication date: 1993-06-29T00:00:00Z
review text: Let \(K\) be an analytic subgroup of a connected, simply connected nilpotent Lie group \(G\). Given a unitary character \(\chi\) of \(K\), the orbit method tells us how to describe the canonical central decomposition of the monomial representation \(\tau = \text{ind}^ G_ K\chi\) [cf. \textit{L. Corwin, P. Greenleaf} and \textit{G. Grélaud}: Trans. Am. Math. Soc. 304, 549-583 (1987; Zbl 0629.22005)]. Now the authors continue their study for the case where \(\tau\) has finite multiplicities. Let \(C^ \infty(G,\tau) = \{\psi\in C^ \infty(G): \psi(kx) = \chi(k)\psi(x)\) for all \(k \in K\), \(g\in C\}\) and consider the algebra \(\mathbb{D}_ \tau(K\setminus G)\) of differential operators on \(G\), taking \(C^ \infty(G,\tau)\) into itself and commuting there with the right action of \(\tau\). Let \(\mathfrak k\) (resp. \(\mathfrak g\)) be the Lie algebra of \(K\) (resp. \(G\)). It follows that there is an \(f\in {\mathfrak g}^*\) such that \(\chi(\text{exp }Y) = e^{2\pi if(Y)}\) for all \(Y\in {\mathfrak k}\). Suppose that \(\tau\) has finite multiplicities. Then the authors showed [in Commun. Pure Appl. Math. 45, 681-748 (1992)] the commutativity of \(\mathbb{D}_ \tau(K\setminus G)\), which was in fact isomorphic to a generating subalgebra of the field \(\mathbb{C}(\Omega_ \tau)^ K\) of \(\text{Ad}^*(K)\)-invariant rational functions on \(\Omega_ \tau = f+{\mathfrak k}^ \perp\subset {\mathfrak g}^*\). In this paper, they assume the existence of a subalgebra which polarizes generic \(\xi \in \Omega_ \tau\) and is normalized by \(\mathfrak k\). This condition enables them to take a suitable Malcev basis of \(\mathfrak g\), and to analyze the corresponding Malcev-Fourier transform in order to get the following main results. \(\mathbb{D}_ \tau(K\setminus G)\) is isomorphic to the algebra \(\mathbb{C}[\Omega_ \tau]^ K\) of \(\text{Ad}^*(K)\)-invariant polynomials on \(\Omega_ \tau\). Furthermore every nonzero element of \(\mathbb{D}_ \tau(K\setminus G)\) has a tempered funda…
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 220667
zbMATH Keywords: simply connected nilpotent Lie group
zbMATH Keywords: orbit method
zbMATH Keywords: central decomposition
zbMATH Keywords: monomial representation
zbMATH Keywords: differential operators
zbMATH Keywords: multiplicities
zbMATH Keywords: commutativity
zbMATH Keywords: rational functions
reviewed by: Hidenori Fujiwara
MaRDI profile type: Publication
cites work: Resolution of Singularities and Division of Distributions
cites work: Analyse harmonique sur les espaces symétriques nilpotents
cites work: Multiplicité un pour les espaces symétriques exponentiels
cites work: Q4092185
cites work: Singular fourier integral operators and representations of nilpotent lie groups
cites work: A canonical approach to multiplicity formulas for induced and restricted representations of nilpotent lie groups
cites work: Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity
cites work: Direct Integral Decompositions and Multiplicities for Induced Representations of Nilpotent Lie Groups
cites work: On some \(C^ *\)-algebras considered by Glimm
cites work: Sur Les Représentations Unitaires Des Groupes De Lie Nilpotents. III
cites work: Q4173774
cites work: Représentations monomiales des groupes de Lie nilpotents. (Monomial representations of nilpotent Lie groups)
cites work: Complex Fourier Analysis on a Nilpotent Lie Group
cites work: Q4188000
cites work: On a connection between nilpotent groups and oscillatory integrals associated to singularities
cites work: Q3965079
cites work: Résolubilité d'équations différentielles et représentation induites. (Solubility of differential equations and induced representations)
cites work: Orbital Parameters for Induced and Restricted Representations
cites work: On the infinitesimal kernel of irreducible representations of nilpotent Lie groups
cites work: Abstract Plancherel theorems and a Frobenius reciprocity theorem
cites work: On \(C^\infty\)-vectors and intertwining bilinear forms for representations of Lie groups
cites work: Unitary representations of solvable Lie groups
cites work: Q5645928
full work available at URL: 0022-1236(92)90030-m
OpenAlex ID: W2075153039
description: scientific article; zbMATH DE number 220667