Riemannian M-spaces with homogeneous geodesics

We investigate homogeneous geodesics in a class of homogeneous spaces called M-spaces, which are defined as follows. Let G/K be a generalized flag manifold with K = C(S) = SxK(1), where S is a torus in a compact simple Lie group G and K-1 is the semisimple part of K. Then, the associated M-space is the homogeneous space G/K-1. These spaces were introduced and studied by H. C. Wang in 1954. We prove that for various classes of M-spaces the only g. o. metric is the standard metric. For other classes of M-spaces we give either necessary, or necessary and sufficient conditions, so that a G-invariant metric on G/K1 is a g. o. metric. The analysis is based on properties of the isotropy representation m = m(1) circle plus center dot center dot center dot circle plus m(s) of the flag manifold G/K [as Ad(K)-modules].