Witt invariants of Weyl groups (Q6612148)

Let \(k_0\) be a field and \(A\) be a functor from the category of field extensions of \(k_0\) into the category of abelian groups (resp. rings) and let \(G\) be a smooth linear algebraic group over \(k_0\). An \(A\)-invariant of \(G\) is a natural transformation \(H^1(-,G)\to A(-)\) of functors \textbf{Fields}\(/k_0\) \(\to\) \textbf{Sets}. The set of all \(A\)-invariants of \(G\) will be denoted \(\hbox{Inv}_{k_0}(G,A)\). It has a natural structure as abelian group (resp. ring) into which \(A(k_0)\) injects as subgroup (resp. subring). In the present paper, the author studies so-called Witt invariants of Weyl groups, i.e., \(\hbox{Inv}_{k_0}(G,W)\), where \(W\) is the functor that maps a field extension \(k\) of \(k_0\) to the Witt ring \(W(k)\) and \(G\) is a Weyl group, where it is assumed that the characteristic of \(k_0\) does not divide the order of \(G\). It is shown that in this situation, if the Weyl group \(G\) is of type \(B_n\), \(C_n\), \(D_n\) or \(G_2\), then \(\hbox{Inv}_{k_0}(G,W)\) is generated as a \(W(k_0)\)-algebra by trace forms and their exterior powers, thus extending a result for type \(A_n\) by [\textit{S. Garibaldi} et al., Cohomological invariants in Galois cohomology. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1159.12311)]. An important ingredient in the proof is the fact that Witt invariants are closely related to cohomological invariants. More precisely, let \(H^n\) denote the Galois cohomology functor \(H^n(-,\mathbb{Z}/2\mathbb{Z})\) and \(I^n\) the functor sending \(k\) to the \(n\)-th power \(I^n(k)\) of the fundamental ideal \(I(k)\) of the Witt ring \(W(k)\). Invoking the Milnor conjectures, there is an exact sequence \N\[\N0\to \hbox{Inv}_{k_0}(G,I^{n+1})\to \hbox{Inv}_{k_0}(G,I^n)\to \hbox{Inv}_{k_0}(G,H^n).\N\]\NFor Weyl groups \(G\) with \(\hbox{char}(k_0)\) not dividing \(|G|\), it is shown that the right-most map is surjective, allowing to conclude that \(\hbox{Inv}_{k_0}(G,W)\) can be generated as \(W(k_0)\)-module by a finite set of lifts of cohomological invariants of \(G\) (Theorem 4.2).\N\NThe paper is well written and also provides an informative account of the basics of the theory of Witt invariants. The author points out that many of the computational techniques developed in the paper can be applied to arbitrary smooth linear algebraic groups.