Closure ordering and the Kostant-Sekiguchi correspondence

Abstract

Let S S be a real semisimple Lie group with Lie algebra s = k + p \mathfrak {s}=\mathfrak {k} +\mathfrak {p} . The Kostant–Sekiguchi correspondence is a bijection between nilpotent S S orbits on s \mathfrak {s} and nilpotent K C K_{\mathbb {C}} orbits on p C \mathfrak {p}_{\mathbb {C}} . In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant–Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.