Projection of orbits and K multiplicities
type: Item
instance of: scholarly article
title: Projection of orbits and K multiplicities
zbMATH Open document ID: 0729.22020
DOI: 10.1016/0022-1236(89)90120-1
author: J. Sengupta
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be a real semisimple noncompact Lie algebra with a chosen Cartan decomposition \({\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}\). Any coadjoint orbit \(\Omega\) in \({\mathfrak g}^*\) carries a canonical invariant measure \(\beta_{\Omega}\). The main result is an explicit formula for the push forward \(J_*\beta_{\Omega}\) under the canonical projection \({\mathfrak g}^*\to {\mathfrak k}^*\) in the case of a regular orbit. The formula is technically complicated; it involves, of course, structural data of the algebra, besides others the root system of a Cartan subalgebra adapted to the chosen Cartan decomposition. The formula is a generalization of a result of \textit{M. Duflo}, \textit{G. Heckman} and \textit{M. Vergne} [in Mem. Soc. Math. Fr., Nouv. Sér. 15, 65-128 (1984; Zbl 0575.22014)] where the case of regular elliptic orbits was treated. At the end of the paper a heuristic interpretation for the formula is given as a ``continuous manifestation'' of the Frobenius reciprocity theorem in the representation theory of compact groups.
reviewed by: D. Poguntke
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 4204694
zbMATH Keywords: Liouville measure
zbMATH Keywords: Hamiltonian space
zbMATH Keywords: moment map
zbMATH Keywords: character formula
zbMATH Keywords: invariant distribution
zbMATH Keywords: K-multiplicities
zbMATH Keywords: real semisimple noncompact Lie algebra
zbMATH Keywords: Cartan decomposition
zbMATH Keywords: coadjoint orbit
zbMATH Keywords: invariant measure
zbMATH Keywords: regular orbit
zbMATH Keywords: root system
zbMATH Keywords: Frobenius reciprocity
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90120-1
OpenAlex ID: W2082110557
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description: scientific article; zbMATH DE number 4204694