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The Grothendieck-Riemann-Roch theorem for group scheme actions

The Grothendieck-Riemann-Roch theorem for group scheme actions
The Grothendieck-Riemann-Roch theorem for group scheme actions
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.

Soit G un groupe ou un schema en groupes. Nous etablissons des formules pour la caracteristique Eulerienne equivariante pour les G-modules localement libres sur un G-schema projectif : nous prouvens le theoreme de Adams-Riemann-Roch et, sous l'hypothese d'une certaine continuite pour l'application image directe, le theoreme de Grothendieck-Riemann-Roch en K-theorie equivariante (superieure). De plus, nous presentons les applications suivantes : le theoreme de Adams-Riemann-Roch implique que les operations de Adams et l'induction pour les representations commutent. Dans le cas d'une variete G/B de drapeaux, l'hypothese de continuite mentionnee ci-dessus est verifiee et le theoreme de Grothendieck-Riemann-Roch apporte alors une nouvelle demonstration de la formule des caracteres de Weyl.
0012-9593
415-458
Koeck, B.
84d11519-7828-43a6-852b-0c1b80edeef9
Koeck, B.
84d11519-7828-43a6-852b-0c1b80edeef9

Koeck, B. (1998) The Grothendieck-Riemann-Roch theorem for group scheme actions. Annales Scientifiques de l'Ecole Normale Supérieure, 31 (3), 415-458. (doi:10.1016/S0012-9593(98)80140-7).

Record type: Article

Abstract

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.

Soit G un groupe ou un schema en groupes. Nous etablissons des formules pour la caracteristique Eulerienne equivariante pour les G-modules localement libres sur un G-schema projectif : nous prouvens le theoreme de Adams-Riemann-Roch et, sous l'hypothese d'une certaine continuite pour l'application image directe, le theoreme de Grothendieck-Riemann-Roch en K-theorie equivariante (superieure). De plus, nous presentons les applications suivantes : le theoreme de Adams-Riemann-Roch implique que les operations de Adams et l'induction pour les representations commutent. Dans le cas d'une variete G/B de drapeaux, l'hypothese de continuite mentionnee ci-dessus est verifiee et le theoreme de Grothendieck-Riemann-Roch apporte alors une nouvelle demonstration de la formule des caracteres de Weyl.

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Published date: May 1998

Identifiers

Local EPrints ID: 29779
URI: https://eprints.soton.ac.uk/id/eprint/29779
ISSN: 0012-9593
PURE UUID: 5706f08c-43a7-4dbb-9326-3f8be022aeb4

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Date deposited: 18 May 2007
Last modified: 17 Jul 2017 15:57

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