Algebraic cycles and completions of equivariant K-theory

Let $G$ be a complex, linear algebraic group acting on an algebraic space $X$. The purpose of this paper is to prove a Riemann-Roch theorem (Theorem 5.3) which gives a description of the completion of the equivariant Grothendieck group $G_0(G,X)$ at any maximal ideal of the representation ring $R(G) \otimes \C$ in terms of equivariant cycles. The main new technique for proving this theorem is our non-abelian completion theorem (Theorem 4.3) for equivariant $K$-theory. Theorem 4.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups.