Support varieties for rational representations

We introduce support varieties for rational representations of a linear algebraic group G of exponential type over an algebraically closed field k of characteristic p > 0. These varieties are closed subspaces of the space V(G) of all 1-parameter subgroups of G. The functor M -> V(G)(M) satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between V(G)(M) and the family of support varieties V-r(G)(M) obtained by restricting the G action to Frobenius kernels G((r)) subset of G. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational G-modules.