Composantes irr\'eductibles de la vari\'et\'e commutante nilpotente d'une alg\`ebre de Lie sym\'etrique semi-simple

TL;DR

This paper investigates the structure of the nilpotent commuting variety in semisimple Lie algebras with involution, proving a conjecture about its irreducible components in various cases.

Contribution

It extends the proof of a conjecture on the irreducible components of the nilpotent commuting variety to new classes of semisimple Lie algebras with involution.

Findings

Confirmed the conjecture for multiple cases beyond the known example

Established equidimensionality of the nilpotent commuting variety in these cases

Identified the indexing of irreducible components by p-distinguished elements

Abstract

Let \theta be an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,\theta) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of p-distinguished elements. This conjecture was established by A. Premet in the case (g \times g, \theta) where \theta(x,y)=(y,x). In this work we prove the conjecture in a significant number of other cases.