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 Livsic-type result to these possibly noncompact and nonaccessible systems. We also prove central limit theorem for Birkhoff averages points horospherical orbit. allows us show that variance is nonzero provided test function mean with respect an invariant probability measure.