  The goal of this paper is to prove a version of the non-abelian localization
theorem for the rational equivariant K-theory of a smooth variety $X$ with the
action of a linear algebraic group $G$. We then use this to prove a
Riemann-Roch theorem which represents the completion of the higher equivariant
K-theory of $X$ at various maximal ideals of the representation ring, in terms
the equivariant higher Chow groups. This generalizes a result of Edidin and
Graham to higher $K$-theory with rational coefficients.
