Degree three cohomological invariants of reductive groups (Q329438)

Let \(G\) be a linear algebraic group defined over a field \(F\). Consider two functorsNEWLINENEWLINE(1) \(G\)-torsors : \({\mathrm{Fields}}_{F} \rightarrow {\mathrm{Sets}},\)NEWLINENEWLINEwhere \({\mathrm{Fields}}_{F}\) denotes the category of fields extensions of \(F.\) The functor takes a field \(K\) to the set of isomorphism classes of \(G\)-torsors over \({\mathrm{Spec}K}.\)NEWLINENEWLINE(2) \( {\Phi} : {\mathrm{Fields}}_{F} \rightarrow {\mathrm{abelian \,\, Groups}}.\)NEWLINENEWLINEThen the \({\Phi}\)-invariant of \(G\) is a morphism of functors \(I: G-{\mathrm{torsors}} \rightarrow {\Phi}\) viewed as functors to Sets. The group of \({\Phi}\)-invariants \({\mathrm{Inv}}(G,{\Phi})\) contains a subgroup of normalized invariants \({\mathrm{Inv}}(G,{\Phi})_{\mathrm{norm}}\) i.e. such that \(I(E)=0\) for every trivial \(G\)-torsor. In the paper the authors consider the cohomology functors \({\Phi}\) taking a field \(K/F\) to the Galois cohomology \(H^{n}(K, {\mathbb Q}/{\mathbb Z}(j))\) and denote by \({\mathrm{Inv}}^{n}(G, {\mathbb Q}/{\mathbb Z}(j))\) the group of cohomological invariants of \(G\) of degree \(n\) with coefficients in \({\mathbb Q}/{\mathbb Z}(j).\) The main result of the paper is the followingNEWLINENEWLINETheorem. Let \(G\) be a split reductive group \(T\subset G\) a split maximal torus, \(W\) the Weyl group and \(C\) the kernel of the universal cover of the commutator subgroup of \(G.\) Then there is an exact sequence NEWLINE\[NEWLINE 0\rightarrow C^{*}{\otimes}F^{\times} \rightarrow {\mathrm{Inv}}^{3}(G, {\mathbb Q}/{\mathbb Z}(2))_{\mathrm{norm}}\rightarrow S^{2}(T^{*})^{W}/{\mathrm{Dec}(G)} \rightarrow 0.NEWLINE\]