On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)

Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993 Helminck , A. G. , Wang , S. P. ( 1993 ). On rationality properties of involutions of reductive groups . Adv. Math. 99 ( 1 ): 26 – 96 .Crossref], Web of Science ®] , Google Scholar]), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.