Convexity of the A-restricted momentum image μ_a(X) in the general smooth compact case

Determine whether the image μ_a(X) ⊂ 𝔞 of the restricted momentum map μ_a := π_𝔞 ∘ μ for the A = exp(𝔞)-action is always convex when X is a connected smooth compact G-invariant submanifold of a Kähler manifold Z, where G ⊂ U^ is a real reductive subgroup compatible with the Cartan decomposition and μ: Z → i𝔲 is a U-equivariant momentum map; equivalently, ascertain whether there exists any connected smooth compact example for which μ_a(X) is non-convex.

Background

In the paper’s general setup, a connected complex reductive group U^ acts holomorphically on a Kähler manifold Z with U Hamiltonian, and a real reductive subgroup G ⊂ U^ compatible with the Cartan decomposition acts on a compact G-invariant submanifold X ⊂ Z. For a maximal abelian subalgebra 𝔞 ⊂ 𝔭, the map μ𝔞 := π𝔞 ∘ μ is the restricted momentum map for A = exp(𝔞). Its convex hull P = conv(μ_𝔞(X)) is a convex polytope.

In the complex case (G complex reductive, X connected), one has P = μ𝔞(X), hence μ𝔞(X) is convex (Atiyah–Guillemin–Sternberg). Beyond the complex case, the authors state they know of no connected smooth compact example where μ𝔞(X) fails to be convex. The open problem is to decide whether μ𝔞(X) is always convex or whether a counterexample exists.