Riemann–Roch for equivariant K-theory

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We show that for a linear algebraic group G acting on a smooth quasi-projective scheme X over a field, there is a Chern character map KiG(X)⊗R(G)R(G)ˆ→chXGCHG⁎(X,i)⊗S(G)S(G)ˆ with rational coefficients, which is an isomorphism. This establishes the equivariant version of the Riemann–Roch isomorphism between the higher algebraic K-theory and the higher Chow groups of smooth quasi-projective schemes.