Approximation of nilpotent orbits for simple Lie groups

TL;DR

This paper studies how continuous families of adjoint orbits in non-compact simple Lie groups approach nilpotent orbits, providing explicit descriptions and approximation methods, especially for special cases like SL(n,R) and SU(p,q).

Contribution

It offers a systematic topological framework for limits of adjoint orbits and explicit descriptions of resulting nilpotent orbits, including approximation techniques.

Findings

Limits of adjoint orbits are finite unions of nilpotent orbits.

Explicit descriptions of nilpotent orbits via Richardson orbits for hyperbolic elements.

Approximation of nilpotent orbits by elliptic semisimple orbits.

Abstract

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 non-compact simple Lie groups. 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.