Adeles and the spectrum of compact nilmanifolds
type: Item
instance of: scholarly article
title: Adeles and the spectrum of compact nilmanifolds
zbMATH Open document ID: 0697.22009
author: Jeffrey Fox
published in: Pacific Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let G be a simply-connected, connected nilpotent Lie group. Let \(\Gamma\) be a discrete cocompact subgroup of G, and let \(\lambda\) be the quasi- regular representation of \(\Gamma\), i.e. the representation of G induced by the trivial one-dimensional representation of \(\Gamma\). Then \(\lambda\) can be decomposed into a direct sum \(\lambda =\sum m(\pi,\lambda)\pi\) of irreducible representations \(\pi\) of G with multiplicities m(\(\pi\),\(\lambda)\). The problem of determining the multiplicities m(\(\pi\),\(\lambda)\) has been initiated by \textit{L. Auslander}, \textit{L. Green}, \textit{F. Hahn} [Flows on homogeneous spaces (Princeton 1963; Zbl 0106.368)] and \textit{C. Moore} [Ann. Math., II. Ser. 82, 146-182 (1965; Zbl 0139.307)]. The problem was then completely solved by \textit{R. Howe} [Am. J. Math. 93, 163-172 (1971; Zbl 0215.118)] and \textit{L. Richardson} [Am. J. Math. 93, 173-190 (1971; Zbl 0265.43012)]. Another formula for m(\(\pi\),\(\lambda)\) was given by \textit{L. Corwin}, \textit{F. Greenleaf} [J. Funct. Anal. 21, 123-154 (1976; Zbl 0319.22011)] some time later. In the present paper the author takes a different approach. He first develops a ``rational'' Kirillov theory for the adele group \(G_{{\mathbb{A}}}\) which he uses to determine the decomposition of \(L^ 2(G_{{\mathbb{A}}}/G_{{\mathbb{Q}}})\). From this he deduces the multiplicity formulas of Howe-Richardson and Corwin-Greenleaf as simple consequences.
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 4142342
zbMATH Keywords: connected nilpotent Lie group
zbMATH Keywords: discrete cocompact subgroup
zbMATH Keywords: quasi-regular representation
zbMATH Keywords: irreducible representations
zbMATH Keywords: adele group
zbMATH Keywords: multiplicity formulas
MaRDI profile type: Publication
full work available at URL: pjm.1989.140.233
OpenAlex ID: W1966428673
DOI: 10.2140/PJM.1989.140.233
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description: scientific article; zbMATH DE number 4142342