The Grothendieck-Riemann-Roch theorem for group scheme actions, by Bernhard Koeck

Let G be a group or a group scheme. We establish formulas for the equivariant Euler characteristic of locally free G-modules on a projective G-scheme X: We prove an Adams- Riemann-Roch theorem and, under a certain continuity assumption for the push-forward map, a Grothendieck-Riemann- Roch theorem in (higher) equivariant algebraic K-theory. Furthermore, we present the following applications: The Adams-Riemann-Roch theorem specializes to an interchanging rule between Adams operations and induction for representations. In case of a flag variety G/B, the above continuity assumption is verified, and the Grothendieck-Riemann-Roch theorem for this situation yields a new proof of the Weyl character formula.

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