Upper motives of outer algebraic groups

TL;DR

This paper studies the motivic structure of projective homogeneous varieties under outer algebraic groups, revealing that all indecomposable summands are shifts of a specific upper summand, extending known results from inner to outer types.

Contribution

It establishes the structure of motivic decompositions for outer type groups, generalizing previous inner type results and identifying the universal form of indecomposable summands.

Findings

Every indecomposable motivic summand is a shift of an upper summand.

The complete motivic decomposition contains a uniquely understandable upper summand.

Results extend known inner type motivic decompositions to outer type groups.

Abstract

Let G be a semisimple affine algebraic group over a field F. Assuming that G becomes of inner type over some finite field extension of F of degree a power of a prime p, we investigate the structure of the Chow motives with coefficients in a finite field of characteristic p of the projective G-homogeneous varieties. The complete motivic decomposition of any such variety contains one specific summand, which is the most understandable among the others and which we call the upper indecomposable summand of the variety. We show that every indecomposable motivic summand of any projective G-homogeneous variety is isomorphic to a shift of the upper summand of some (other) projective G-homogeneous variety. This result is already known (and has applications) in the case of G of inner type and is new for G of outer type (over F).