Central Limit theorem and cohomological equation on homogeneous spaces

The dynamics of one parameter diagonal group actions on finite volume homogeneous spaces has a partially hyperbolic feature. In this paper we extend the Liv\v{s}ic type result to these possibly noncompact and nonaccessible systems. We also prove a central limit theorem for the Birkhoff averages of points on a horospherical orbit. The Liv\v{s}ic type result allows us to show that the variance of the central limit theorem is nonzero provided that the test function has nonzero mean with respect to an invariant probability measure.