Smallest Complex Nilpotent Orbits with Real Points

Let g be a non-compact real simple Lie algebra without complex structure, and denote by g(C) the complexification of g. This paper focuses on non-zero nilpotent adjoint orbits in g(C) meeting g. We show that the poset consisting of such nilpotent orbits equipped with the closure ordering has the minimum O-min,g(GC). Furthermore, we determine such O-min,g(GC) in terms of the Dynkin-Kostant classification even in the cases where O-min,g(GC) does not coincide with the minimal nilpotent orbit in g(C). We also prove that the intersection O-min,g(GC) boolean AND g is the union of all minimal nilpotent orbits in g.