Metric operator and geodesic orbit property for a standard homogeneous Finsler metric

In this paper, we introduce the metric operator for a compact homogeneous Finsler space, and use it to investigate the geodesic orbit property. We define the notion of standard homogeneous (₁, , ₛ) -metric which generalizes the notion of standard homogeneous (₁, ₂) -metric. We classify all connected simply connected homogeneous manifold G/H with a compact connected simple Lie group G and two irreducible summands in its isotropy representation, such that there exists a standard homogeneous (₁, ₂) -metric which is g. o. but not naturally reductive on G/H. We also prove that on a generalized Wallach space which is not a product of three symmetric spaces, any standard homogeneous (₁, ₂, ₃) -metric F with respect to the canonical decomposition is g. o. on G/H if and only if F is a normal homogeneous Riemannian metric.