Hamiltonian reductions of free particles under polar actions of compact Lie groups

Classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds are investigated. The reduced systems are described under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifolds meeting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of spin Calogero-Sutherland type.

💡 Research Summary

The paper develops a unified framework for performing Hamiltonian reduction of free geodesic motion on complete Riemannian manifolds when the symmetry group is a compact Lie group acting polarly. A polar action means that there exists a closed, connected, regularly embedded submanifold (a section) that meets every group orbit orthogonally and serves as a global transversal. Under this geometric hypothesis the authors are able to carry out both classical and quantum reductions in a completely explicit way.

The starting point is a free particle on a Riemannian manifold ((M,g)) with Hamiltonian (H=\frac12|p|^{2}). The compact Lie group (G) acts smoothly and isometrically on (M). The associated momentum map (\mu:T^{}M\to\mathfrak g^{}) encodes the conserved quantities. For a regular value (\xi) of (\mu) the Marsden–Weinstein reduction yields the reduced phase space (\mathcal P_{\xi}= \mu^{-1}(\xi)/G_{\xi}). Because the action is polar, the section (\Sigma) provides a diffeomorphism (\mathcal P_{\xi}\cong T^{*}\Sigma\times\mathcal O_{\xi}), where (\mathcal O_{\xi}) is the coadjoint orbit through (\xi). Consequently the reduced dynamics splits into two parts: (i) the “external” motion on the section, described by coordinates (x\in\Sigma) and momenta (p_{x}); (ii) an internal “spin” degree of freedom living on the orbit (\mathcal O_{\xi}). The reduced Hamiltonian contains the kinetic term on (\Sigma) plus a potential that couples the external coordinates to the spin variables through the root system of the group.

Quantization proceeds by restricting the original Hilbert space (L^{2}(M)) to the subspace of (G)-invariant wave functions and applying the quantum analogue of Marsden–Weinstein reduction (the Guillemin–Sternberg “quantization commutes with reduction” principle). The reduced Hilbert space is isomorphic to (L^{2}(\Sigma)\otimes V_{\xi}), where (V_{\xi}) carries an irreducible representation of the stabilizer (G_{\xi}). The quantum Hamiltonian acts as a differential operator on (\Sigma) together with matrix-valued spin operators originating from the representation of (\mathfrak g_{\xi}). Thus the quantum reduction reproduces the classical spin‑Calogero–Sutherland structure at the operator level.

A particularly rich class of examples arises when the polar action is hyperpolar, i.e. the section is flat. This situation occurs for the adjoint action of a compact Lie group on itself and for the isotropy actions on symmetric spaces (G/K). In the adjoint case the section can be identified with a maximal torus (T) and its Lie algebra (\mathfrak t). The reduced system becomes a spin‑Calogero–Moser model with potential \