Polar actions on Hilbert space
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 \(G\).
isometric \(H\)-action on a Riemannian manifold
scientific article; zbMATH DE number 734769
scientific article