Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces

Abstract

Abstract. We obtain the full classification of coisotropic and polar isometric actions of compact Lie groups on irreducible Hermitian symmetric spaces.

Polar actions on symmetric spaces

“…This approach was further pursued by Biliotti and Gori [3], who classified coisotropic and polar actions on the complex Grassmannians G k (C n ). The classification of coisotropic actions on the compact irreducible Hermitian symmetric spaces was recently completed by Biliotti [2], showing in particular that polar actions on these spaces are hyperpolar, which led Biliotti to conjecture that this holds for all compact irreducible symmetric spaces.…”

“…Let us consider the SU(4) ⊗ SU(2)-action on SU(8)/Sp(4), a slice representation is Ad SO(4) ⊗ Ad SU (2) , which is non-polar [1] and polarity minimal, hence we may apply Lemma 6.4 (ii). For the Sp(3) ⊗ Sp(1)-action we find a slice representation P 2 (Sp(3)) ⊗ R 2 of Sp(3) ⊗ U(1), which is also non-polar [13] and polarity minimal, hence Lemma 6.4 (ii) also applies in this case.…”

“…Subactions of Spin(9) on G 3 (R 16 ). The action on G 3 (R 16 ) was shown not to be polar in [33], the slice representation being equivalent to a 16-dimensional non-polar irreducible representation of Sp(1)•Sp (2). Thus by Lemma 6.4 (ii), no subaction of the Spin(9)-action on G 3 (R 16 ) is polar.…”

From isoparametric submanifolds to polar foliations

“…Many special cases of Theorem 5.4 had been proved previously, most notably under the assumption that the symmetric space is Hermitian in [2].…”

Hamiltonian actions and homogeneous Lagrangian submanifolds

“…(T x Gx) ⊥ø ⊂ T x Gx. Coisotropic actions are intensively studied in [17] and [14] and in the following paper [26], [6], [7] the complete classification of compact connected Lie groups acting isometrically and in a Hamiltonian fashion on irreducible compact Hermitian symmetric spaces are given. In a forthcoming paper [8], we shall study coisotropic actions of Lie groups acting properly and in a Hamiltonian fashion on a symplectic manifold M. An equivalent condition for a connected Lie group G acts coisotropically on M is that for every α ∈ g * the set Gµ −1 (α)/G are points [8].…”