Classification of K_F-orbits of Unipotent Elements in Symmetric F-varieties of SL(n, F)

Abstract

Richardson proved in 1982 that, given an algebraic group G and some involution, we could have only a finite number of K-orbits of unipotent elements in the symmetric variety P = G/K over an algebraically closed field, where K is the fixed point group of the involution. A question arises naturally: what if the field is not algebraically closed? In this thesis we try to answer this question and go a little further by listing all K_F-orbits of unipotent elements in P explicitly. We work on the symmetric F-variety P = G_F/K_F for the special linear group over an arbitrary field F of characteristic not 2. We classify all K_F-orbits of unipotent elements in P for all inner involutions for the special linear group. For Cartan (outer) involution, we classify K-orbits for small n only and illustrate how to get the canonical form for general n. Further proofs are still needed. We also classify G_F-orbits of unipotent elements in G_F.