Approximation of nilpotent orbits for simple Lie groups

We propose a systematic and topological study of limits \(\lim_{\nu\to 0^+}G_\mathbb{R}\cdot(\nu x)\) of continuous families of adjoint orbits for a non-compact simple real Lie group \(G_\mathbb{R}\). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of \(\mathrm{SL}_n(\mathbb{R})\) and \(\mathrm{SU}(p,q)\) are computed in detail.