Completions of Higher Equivariant K-theory

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.