A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces
type: Item
instance of: scholarly article
title: A commutativity criterion for certain algebras of invariant differential operators on nilpotent homogeneous spaces
zbMATH Open document ID: 1046.43002
published in: Mathematische Annalen
publication date: 2004-07-07T00:00:00Z
review text: Let \(G\) be a connected, simply connected real nilpotent Lie group with Lie algebra \( \mathfrak g\), \( H\) a connected closed subgroup of \(G\) with Lie algebra \(\mathfrak h\) and \(f\) a linear form on \( \mathfrak g\) satisfying \(\langle f, [\mathfrak h, \mathfrak h] \rangle = (0)\). Let \(\chi_f\) be the unitary character of \(H\) defined by \( \chi_f( \exp X)= e^{if(X)},\;X\in \mathfrak h.\) Consider the unitary representation \( \tau = \text{Ind}_H^G\chi_f\) induced from the character \(\chi_f\) of \(H\). So, \(\tau \) decomposes into a continuous sum of unitary irreducible representations of \(G\), \[ \tau = \int ^\oplus _{\widehat {G}} m(\pi) \pi d\mu(\pi) \] where \( m(\pi)\) denotes the multiplicity of \( \pi\) and \( \mu\) a Plancherel measure of \(\tau \) on the unitary dual \(\widehat G\) of \(G\). The following fact is well known: either the multiplicity function \(m(\pi)\) is finite and uniformly bounded for \(\mu\)-almost all \(\pi \in \widehat G\), or it is infinite [cf. \textit{L. Corwin}, \textit{F. P. Greenleaf} and \textit{G. Grelaud}, Trans. Am. Math. Soc. 304, 549--583 (1987; Zbl 0629.22005); \textit{R. Lipsman}, Trans. Am. Math. Soc. 313, 433--473 (1989; Zbl 0683.22009)]. The paper under review deals with the algebra \(D_\tau (G/H)\) of linear \( C^\infty\)-differential operators keeping invariant the space \(C^\infty(G,H,f)\) of \(C^\infty \) complex functions \(\phi\) on the group \(G\) and satisfying the cocycle covariance relation \(\phi(gh)= \chi_f(h^{-1}) \phi(g)\) for all \(h\in H\) and \(g\in G\), and commuting with left translations of \(G\) on that space. \textit{L. Corwin} and \textit{F. P. Greenleaf} proved that under the assumption that \(\tau\) is of finite multiplicities the algebra \(D_\tau(G/H)\) is commutative [Commun. Pure Appl. Math. 45, 681--748 (1992; Zbl 0812.43004)] and conjectured later that the converse is also true. In this interesting paper, the authors prove this conjecture for an arbitrary connected simply co…
reviewed by: Ali Baklouti
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E27
zbMATH DE Number: 2078342
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: differential operator
zbMATH Keywords: induced representation
author: Hidenori Fujiwara
author: Gérard Lion
author: Bernard Magneron
author: Salah Mehdi
MaRDI profile type: Publication
full work available at URL: s00208-003-0464-3
OpenAlex ID: W1975414122
DOI: 10.1007/S00208-003-0464-3
description: scientific article; zbMATH DE number 2078342