Riemann-roch for Equivariant K-theory

The goal of this paper is to prove the equivariant version of Bloch’s Riemann-Roch isomorphism between the higher algebraic K-theory and the higher Chow groups of smooth varieties. We show that for a linear algebraic group G acting on a smooth variety X , although there is no Chern character map from the equivariant K-groups to equivariant higher Chow groups, there is indeed such a map K i (X)⊗R(G)R̂(G) ch −→ CH∗ G (X, i)⊗S(G)Ŝ(G) with rational coefficients, which is an isomorphism. This implies the Riemann-Roch isomorphism K̂ i (X) b τ G X −−→ ̂ CH∗ G (X, i). The case i = 0 provides a stronger form of the Riemann-Roch theorem of Edidin and Graham (cf. [6]).