Multivariate random fields on some homogeneous spaces

The generalized continuous random fields of second order with values in arbitrary complex normed space $X$ in the case when their arguments belong to homogeneous space with compact transformation group $G$ are considered. Such fields are harmonizable in some sense. The spectral representations of homogeneous random fields in $X$ and $G$-invariant positive definite operator-valued kernels are obtained. The special case of random fields with values in complex Hilbert space and random fields on three-dimensional spheres are also studied.