### 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.

Bernhard Koeck <bk@ma2s2.mathematik.uni-karlsruhe.de>