Polar actions on Hilbert space
type: Item
instance of: scholarly article
title: Polar actions on Hilbert space
zbMATH Open document ID: 0818.57023
DOI: 10.1007/BF02926445
published in: The Journal of Geometric Analysis
publication date: 1995-08-22T00:00:00Z
review text: An isometric \(H\)-action on a Riemannian manifold \(X\) is called polar if there exists a closed submanifold \(S\) of \(X\) that meets every \(H\)-orbit and always meets orbits orthogonally \((S\) is called a section). Let \(G\) be a compact Lie group equipped with a bi-invariant metric, \(H\) a closed subgroup of \(G \times G\), and let \(H\) act on \(G\) isometrically by \((h_ 1, h_ 2) \cdot x = h_ 1 xh_ 2^{-1}\). Let \(P(G,H)\) denote the group of \(H^ 1\)-paths \(g : [0,1] \to G\) such that \((g(0), g(1)) \in H\), and let \(P(G,H)\) act on the Hilbert space \(V = H^ 0([0,1], {\mathfrak g})\) isometrically by \(g*u = gug^{-1} - g'g^{-1}\). We prove that if the action of \(H\) on \(G\) is polar with a flat section then the action of \(P(G,H)\) on \(V\) is polar. Principal orbits of polar actions on \(V\) are isoparametric submanifolds of \(V\), and are infinite dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples of \(P(G,H)\)-actions for suitable choice of \(H\) and \(G\).
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 734769
zbMATH Keywords: isometric \(H\)-action on a Riemannian manifold
zbMATH Keywords: polar
zbMATH Keywords: section
zbMATH Keywords: compact Lie group
zbMATH Keywords: polar actions
zbMATH Keywords: isoparametric submanifolds
zbMATH Keywords: flag manifolds
zbMATH Keywords: affine Kac-Moody groups
author: Chuu-Lian Terng
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 734769