Note on rational points in nilpotent orbits of semisimple groups (Q1279766)

The author considers the problem of the existence of rational points on nilpotent orbits (1985, Carter; 1992, Collingwood, McGovern) of a connected semisimple linear algebraic \(k\)-group \(G\) (1991, Borel; 1994, Springer) defined over a field \(k\) with characteristic 0 and identified here with the group \(G(\overline k)\) of \(\overline k\)-points of \(G\) (\(\overline k\) being an algebraic closure of \(k\)). The group \(G_k\) of \(k\)-rational points of \(G\) and the related Lie algebra \(g_k\) are also introduced. Necessary and sufficient conditions are determined and proved for the nonzero nilpotent \(G(\overline k)\)-orbit \(O\) belonging to the Lie algebra \(g(\overline k)\) of \(G\) to be defined over the field \(k\) (Theorem 1) as well as to have a \(k\)-rational point, i.e., \(O\cap g_k\neq 0\) (Theorem 2).