Relative equivariant motives and modules

TL;DR

This paper explores the relationship between equivariant motives of flag varieties and parabolic modules, revealing how motivic decompositions depend on module decompositions, using advanced localization and nilpotence techniques.

Contribution

It introduces categories of equivariant motives linked to parabolic modules and establishes their connection, providing new insights into motivic decompositions of flag varieties.

Findings

Motivic decomposition type depends on parabolic module decomposition.

New proofs of indecomposable Chow motives for versal flag varieties.

Application of localization and Rost nilpotence principles in this context.

Abstract

We introduce and study various categories of (equivariant) motives of (versal) flag varieties. We relate these categories with certain categories of parabolic (Demazure) modules. We show that the motivic decomposition type of a versal flag variety depends on the direct sum decomposition type of the parabolic module. To do this we use localization techniques of Kostant-Kumar in the context of generalized oriented cohomology as well as the Rost nilpotence principle for algebraic cobordism and its generic version. As an application, we obtain new proofs and examples of indecomposable Chow motives of versal flag varieties.