Tits construction and Rost invariant

TL;DR

This paper proves a Springer type theorem for certain parabolic subgroups in all groups of type E6, utilizing the Rost invariant, Tits construction, and symmetric spaces, advancing understanding of projective homogeneous varieties of exceptional type.

Contribution

It provides the first example of a Springer theorem for projective homogeneous varieties of type ^2E6 beyond Borel subgroups, combining multiple advanced mathematical tools.

Findings

Springer theorem established for type E6 parabolic varieties

First such theorem for ^2E6 varieties beyond Borel subgroups

Uses Rost invariant, Tits construction, and symmetric spaces

Abstract

We show a Springer type theorem for the variety of parabolic subgroups of type $1,2,6$ for all groups of type $E_6$. As far as we know this gives the first example for the validity of the Springer theorem for projective homogeneous varieties of type $^2E_6$ different from varieties of Borel subgroups. The proof combines several topics, notably the Rost invariant, a Tits construction, Cartan's symmetric spaces and indirectly the structure of the Chow motives of projective homogeneous varieties of exceptional type.