
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.
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