An important class of sampled-data systems becomes time-invariant when the systems are "lifted" into an appropriate domain. The purpose of the present paper is to study some basic properties of systems represented by state space equations in the lifted domain. Hidden modes, invariant zeros, and Riccati equations are investigated, and the connection between representations in continuous and lifted domains are clarified. The material treated here is important for solving optimal control problems of generalized sampled-data systems.