On the classification of \(k\)-involutions (Q1576601)

Let \(G\) be a connected reductive algebraic group defined over a field \(k\) of characteristic not \(2\), \(\theta\) an involution of \(G\) defined over \(k\), \(H\) a \(k\)-open subgroup of the fixed point group of \(\theta\), and \(G_k\) and \(H_k\) the set of \(k\)-rational points of \(G\) and \(H\), respectively. The variety \(G_k/H_k\) is called a symmetric \(k\)-variety. The author gives a characterization of the isomorphy classes of symmetric \(k\)-varieties together with their fine structure of restricted root systems. This restricted root system is the root system of a maximal \((\theta,k)\)-split torus of \(G\), i.e., a torus that is both \(\theta\)-split and \(k\)-split. The author gives a characterization of \((\Gamma,\theta)\)-indices, where \(\Gamma\) is the Galois group of a splitting extension of the maximal torus, and a classification of these for \(k\) the real numbers, the \(p\)-adic numbers, finite fields, and number fields.