Minimal parabolic k-subgroups acting on symmetric k-varieties corresponding to k-split groups

Abstract

Symmetric [Formula: see text]-varieties are a natural generalization of symmetric spaces to general fields [Formula: see text]. We study the action of minimal parabolic [Formula: see text]-subgroups on symmetric [Formula: see text]-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of [Formula: see text]-split groups that depends only on the dimension of a maximal [Formula: see text]-split torus contained within the fixed point group of the involution defining the symmetric [Formula: see text]-variety.

Geometry of symmetric spaces of type EIII

“…It is important that one can develop symmetric spaces not only over R or C, but over any field of characteristic not two. In this case they are related to involutions on simple algebraic groups, see the papers by Helminck,Richardson,Springer ([6,11,13] respectively) and Hutchens and Hunnel ([7,8]).…”