T-equivariant motives of flag varieties

Yaylali, Can

doi:10.2140/agt.2025.25.2343

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$T$-equivariant motives of flag varieties

Can Yaylali

Algebraic & Geometric Topology 25 (2025) 2343–2367

DOI: 10.2140/agt.2025.25.2343

Abstract

We use the construction of the stable homotopy category by Khan and Ravi to calculate the integral $T$ -equivariant $K$ -theory spectrum of a flag variety over an affine scheme, where $T$ is a split torus associated to the flag variety. More precisely, we show that the $T$ -equivariant $K$ -theory ring spectrum of a flag variety is decomposed into a direct sum of $K$ -theory spectra of the classifying stack $B T$ indexed by the associated Weyl group. We also explain how to relate these results to the motivic world and deduce classical results for $T$ -equivariant intersection theory and $K$ -theory of flag varieties.

For this purpose, we analyze the motive of schemes stratified by affine spaces with group action, that preserves these stratifications. We work with cohomology theories, that satisfy certain vanishing conditions, which are satisfied for example by motivic cohomology and $K$ -theory.

Keywords

flag varieties, torus actions, $K$-theory, motives, equivariant homotopy theory

Mathematical Subject Classification

Primary: 14F42

Secondary: 14A20, 14L30, 14M15, 19E08

References

Publication

Received: 31 August 2023

Revised: 1 February 2024

Accepted: 14 March 2024

Published: 11 August 2025

Authors

Can Yaylali
TU Darmstadt Darmstadt Germany

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Volume 25, issue 4 (2025)