Asymptotic cone of semisimple orbits for symmetric pairs

TL;DR

This paper extends the understanding of asymptotic cones of semisimple orbits from complex reductive groups to symmetric pairs, revealing new geometric properties related to Richardson nilpotent orbits.

Contribution

It proves an analogue of Borho and Kraft's result for semisimple orbits within symmetric pairs, broadening the scope of geometric orbit analysis.

Findings

Asymptotic cone of semisimple orbits in symmetric pairs is characterized similarly to the classical case.

Establishes a connection between the asymptotic cone and Richardson nilpotent orbits in the symmetric pair setting.

Provides new insights into the geometric structure of orbits in symmetric pairs.

Abstract

Let G be a reductive algebraic group over the complex field and O_h be a closed adjoint orbit through a semisimple element h. By a result of Borho and Kraft (1979), it is known that the asymptotic cone of the orbit O_h is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer Z_G(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair (G, K).