$J$-invariant of linear algebraic groups of outer type

TL;DR

This paper generalizes the $J$-invariant to all semisimple linear algebraic groups, offering decompositions of Chow motives and explicit formulas, thereby advancing understanding of algebraic group invariants.

Contribution

It introduces a comprehensive extension of the $J$-invariant to arbitrary semisimple groups and provides explicit decompositions and formulas for associated Chow motives.

Findings

Complete decompositions of normed Chow motives for all generically quasi-split twisted flag varieties.

Identification of combinatorial patterns in normed Chow groups and motives.

Explicit formulas for the values of the $J$-invariant.

Abstract

We extend the notion of the $J$-invariant to arbitrary semisimple linear algebraic groups and provide complete decompositions for the normed Chow motives of all generically quasi-split twisted flag varieties. Besides, we establish some combinatorial patterns for normed Chow groups and motives and provide some explicit formulae for values of the $J$-invariant.