On Rationality Properties of Involutions of Reductive Groups

Introduction. Let k be a field of characteristic not two and G a connected linear reductive k-group. By a k-involution θ of G, we mean a k-automorphism θ of G of order two. For k = R, C or an algebraically closed field, such involutions have been extensively studied emerging from different interests. As manifested in [8, 18, 28], the interactions with the representation theory of reductive groups are most rewarding. The application of discrete series of affine symmetric spaces to the cohomology of arithmetic subgroups [27] invites the study of Q-involutions. In the present paper, we give a treatment on rationality problems of general k-involutions. Here we generalize most of the earlier results [15, 16, 23, 29], sharpen some and add new ones. Let H be an open subgroup of the fixed point group G θ of an involution θ of G. In