Irregular locus of the commuting variety of reductive symmetric Lie algebras and rigid pairs

TL;DR

This paper investigates the irregular locus of the commuting variety in reductive symmetric Lie algebras, extending previous results and introducing the concept of rigid pairs to analyze reducibility of these varieties.

Contribution

It extends the description of the irregular locus to the symmetric Lie algebra setting and introduces rigid pairs to identify cases of reducibility.

Findings

Describes the irregular locus of the symmetric commuting variety.

Identifies rigid pairs that characterize reducibility.

Provides bounds on the codimension of the irregular locus.

Abstract

The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain some results on the irregular locus of the component of maximal dimension of the "symmetric commuting variety". As a by-product, we study some pairs of commuting elements specific to the symmetric case, that we call rigid pairs. These pairs allow us to find all symmetric Lie algebras whose commuting variety is reducible.