On the classification of polar representations

Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

“…Note also that the reducible actions arising from Tables IIa and IIb are not polar. This can be easily deduced as an application of Theorem 2 (page 313) of [4], while (see [10] and [18]) in the irreducible case we know that u(m) on Sp(m)/U(m), u(m) and su(m) when m is odd on SO(2m)/U(m), Spin(10) and T 1 · Spin(10) on…”

“…Then the K-action is coisotropic if and only if the slice representation is coisotropic (see [15], page 274 Once we have determined the complete list of coisotropic actions on compact irreducible Hermitian symmetric spaces we also have investigated which ones are polar. Dadok [7], Heintze and Eschenburg [10] have classified the irreducible polar linear representations, while I. Bergmann [4] has found all the reducible ones. Using their results we determine in section 7 the complete classification of the polar actions on the following Hermitian symmetric spaces…”

“…Then l = z + sp(2) acts coisotropically, and the scalar cannot be removed. Note that, since the slice of the orbit through 1 ∈ C is R ⊕ C 5 on which SO(5) acts on C 5 , one may prove (see also [10]) that the slice fails to be polar. This case is maximal, since for every h ⊆ sp(2) we have that z + h does not satisfy the dimensional condition.…”

“…Note that the first scalar acts on C while the centralizer of Spin (10) in Spin (12) does not. Hence, the action is multiplicity free, since the Spin(10)−action on C 16 is multiplicity free.…”

Cohomogeneity one hypersurfaces of Euclidean Spaces

“…We also point out that closed subgroups of SO(n) that act transitively on the sphere are completely classified (cf. [7], p. 392).…”