Invariant Kähler polarizations and geometry of vector fields

Ihor Mykytyuk

Applied Mathematics Department, State University ”Lviv Polytechnica“, S. Bandera St., 12, 79013, Lviv, UKRAINE, e-mail: viva@iapmm.lviv.ua

Let G/K be a Riemannian symmetric space. Let 𝔤 = 𝔨 ⊕ 𝔪 be the corresponding invariant splitting of the Lie algebra 𝔤 of G. Geometric constructions which come from geometric quantization naturally lead to complex structures defined on the punctured cotangent bundle T_{0}^{∗}(G/K) = T^{∗}(G/K) − {zero section}. Such structure J_{S} for the spheres was found by Souriau. Later it was observed by Rawnsley, that the length function is strictly plurisubharmonic with respect to the above complex structure J_{S} and defines the Kähler metric on T_{0}^{∗}S^{n} with the canonical symplectic form Ω as the Kähler form. He also observed that J_{S} is invariant with respect to the Hamiltonian flow of the length function (the normalized geodesic flow) and used the Kähler polarization J_{S} to quantize the geodesic flow on the spheres.

Our purpose is to describe G-invariant Kähler structures on T^{∗}(G/K). Denote by ⟨, ⟩ the scalar product on 𝔪 defining the Riemannian metric on G/K. Let S^{+}(𝔪^{ℂ}) be the set of all invertible symmetric (with respect to ⟨, ⟩) operators s : 𝔪^{ℂ} → 𝔪^{ℂ} such that Re s > 0 (positive definite).

Theorem.There is one-to-one correspondence between the set of all Ginvariant Kähler structures {(J, Ω)} on T^{∗}(G/K) and the set {P } of all smooth mappings P : 𝔪 → S^{+}(𝔪^{ℂ}), w ↦ P_{w} such that 1) (Ad_{k} P Ad_{k^{−1}})|𝔪 = P_{Ad_{k} w}, ∀w ∈ 𝔪, k ∈ K; 2) [P_{w}(ξ), P_{w}(η)](w) = −[w, [ξ, η]] on 𝔪, ∀ξ, η ∈ 𝔪, where we consider the maps w ↦ P_{w}(ξ), w ↦ P_{w}(η) as vector fields on 𝔪.

It is described all mappings P satisfying conditions of Theorem in the following cases: (1) if rank G/K = 1, dim G/K ≥ 3; (2) if the corresponding Kähler structure (J, Ω) is invariant under the normalized geodesic flow. For Hermitian symmetric spaces G/K of rank ≥ 2 a family of Kähler structures (of mappings P) is constructed. As an application of these results we obtain 1) a description of the curvature tensor of the symmetric space F_{4}/Spin(9) in terms of its Spin(9)-structure; 2) new results concerning the hyper-Kähler metric on T^{∗}(G/K) if G/K is a Hermitian symmetric space of rank ≥ 1 (in this case the natural holomorphic-symplectic structure underlies the hyper-Kähler metric, whose restriction to G/K is the given homogeneous metric).