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

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.