Hamiltonian minimality of normal bundles over the isoparametric submanifolds

Let N be a complex flag manifold of a compact semi-simple Lie group G, which is standardly embedded in the Lie algebra g of G as a principal orbit of the adjoint action. We show that the normal bundle of N in g is a Hamiltonian minimal Lagrangian submanifold in the tangent space Tg which is naturally regarded as the complex Euclidean space. Moreover, we specify the complex flag manifolds with this property in the class of full irreducible isoparametric submanifolds in the Euclidean space.