Invariant geodesic orbit metrics on homogeneous spaces with intermediate subgroups

In this paper, we study G -g.o. metrics on compact homogeneous spaces G / H with an intermediate subgroup K such that H ⊂ K ⊂ G . In the beginning, we prove that the restricted metrics of g on K / H and G / K are both g.o. metrics under certain conditions when g is a G -g.o. metric on G / H . Then we develop several methods to determine G -g.o. metrics on G / H by the representations of K / H and G / K . As an application, we study g.o. metrics on a class of homogeneous spaces and find that SO ( 11 ) / ( Spin ( 7 ) × SO ( 2 ) ) admits non-naturally reductive g.o. metrics.