Twisted Witt Groups of Flag Varieties

TL;DR

This paper characterizes when twisted Witt groups of flag varieties vanish over algebraically closed fields, revealing new cases and describing their structure when non-zero, using Tate cohomology of representation rings.

Contribution

It provides a complete characterization of the vanishing of twisted Witt groups for flag varieties over algebraically closed fields and describes their module structure in non-zero cases.

Findings

Twisted Witt groups vanish in many previously unknown cases.

When non-zero, the twisted Witt group is a rank-one free module over the untwisted group.

Verification reduces to computing twisted Tate cohomology of representation rings.

Abstract

Calm\`es and Fasel have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, we sharpen their result to an if-and-only-if statement. In particular, we show that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, we find that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading. Our proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their K-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.