On the Tits p-indexes of semisimple algebraic groups

TL;DR

This paper provides a comprehensive description of Tits p-indexes for semisimple algebraic groups over fields, linking algebraic, cohomological, and motivic invariants to classify these groups.

Contribution

It offers a complete characterization of Tits p-indexes over fields and derives criteria for motivic equivalence among semisimple groups, connecting various algebraic and geometric invariants.

Findings

Complete description of Tits p-indexes over fields

Criteria for motivic equivalence of semisimple groups

Establishment of a dictionary linking algebraic, cohomological, and motivic invariants

Abstract

The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria of motivic equivalence for semisimple groups of many types, hence giving a dictionary between classic algebraic structures, representation theory, cohomological invariants and Chow motives of the twisted flag varieties for those groups.