Constructing coherently G-invariant modules

Let G be a reductive group acting on a path algebra kQ as automorphisms. We assume that G admits a graded polynomial representation theory, and the action is polynomial. We describe the quiver QG of the smash product algebra kQ#k[MG]⁎, where MG is the associated algebraic monoid of G. We use QG-representations to construct coherently G-invariant modules of Q. As an application, we construct algebraic semi-invariants on the quiver representation spaces from those G-invariant modules.