A remark on local-global principles for conjugacy classes (Q1810346)

Let \(G\) be a linear algebraic group over a (perfect) field \(k\). We say that \(x, y\in G(k)\) are conjugate in \(G(k)\) (and denoted by \(x\sim y\)) if there is a \(g\in G(k)\) with \(\text{Int}(g)x=gxg^{-1}=y\). If \(x\sim y\) in \(G(k)\), then it implies that \(x\sim y\) in \(G(k_{\xi})\) for each \(\xi\in\text{Spec}\,k\). We say that the (strong) real closures principle holds for the conjugacy class of \(y\) in \(G(k)\) if the converse holds, namely, for \(x,y \in G(k)\), if \(x\sim y\) in \(G(k_{\xi})\) for all \(\xi\) in (any dense subset of) \(\text{Spec}\, k\), implies that \(x\sim y\) in \(G(k)\). This is a local-global principles for the conjugacy classes of \(G\). The main result of this paper is the following Theorem. Let \(G\) be a linear algebraic group over a field \(k\) with an ordering (thus \(\text{Spec} \,k\neq \emptyset\), hence \(\text{char}(k)=0\)) for which the (strong) real closures principle holds. Fix \(y\in G(k)\). The (strong) real closures principle holds for the conjugacy class of \(y\) in \(G(k)\) if and only if the (strong) real closures principle holds in the centralizer \(G_y=Z_G(y)\) of \(y\) in \(G\). Several examples are discussed, e.g., a local-global principles holds for all conjugacy classes of any inner form of \(\text{GL}(n),\, \text{SL}(n),\, \text{SU}(n)\), and for all semisimple conjugacy classes in any inner form of \(\text{Sp}(n)\), over fields \(k\) with virtual cohomological dimension \(\text{vcd}(k)\leq 1\). Over number fields \(k\) (so \(\text{vcd}(k)=2\)), the real closures principle is shown to hold for conjugacy classes in \(\text{GL}(n,k)\), but does not hold for all semisimple conjugacy classes in the unitary, symplectic or orthogonal groups.