Étale representations for reductive algebraic groups with one-dimensional center

A complex vector space V is a prehomogeneous G-module if G acts rationally on V with a Zariski-open orbit. The module is called étale if dim⁡V=dim⁡G. We study étale modules for reductive algebraic groups G with one-dimensional center. For such G, even though every étale module is a regular prehomogeneous module, its irreducible submodules have to be non-regular. For these non-regular prehomogeneous modules, we determine some strong constraints on the ranks of their simple factors. This allows us to show that there do not exist étale modules for G=GL1×S×⋯×S, with S simple.