Algebraic cycles and completions of equivariant $K$-theory

Abstract

Let $G$ be a complex, linear algebraic group acting on an algebraic space $X$. The purpose of this article is to prove a Riemann-Roch theorem (Theorem 6.5) that gives a description of the completion of the equivariant Grothendieck group $G_0(G,X) \otimes {\mathbb C}$ at any maximal ideal of the representation ring $R(G) \otimes {\mathbb C}$ in terms of equivariant cycles. The main new technique for proving this theorem is our nonabelian completion theorem (Theorem 5.3) for equivariant $K$-theory. Theorem 5.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups