On Panyushev's rootlets for infinitesimal symmetric spaces

Let g be a simple Lie algebra, b a fixed Borel subalgebra, R the corresponding root system and W the associated Weyl group. In the first part of this work we decompose the poset of abelian ideals of b into peculiar subposets, which turn out to be isomorphic to right coset representatives obtained from specific subgroups of W. We use this to give a new proof of the Panyushev's one-to-one correspondence between maximal abelian ideals of b and long simple roots of R. In the second part of the work we extend the study to the case where g = g_0 + g_1 is a Z_2-graded Lie algebra. We study the poset of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0, and we decompose it, in the semisimple cases, into specific subposets, that we show are isomorphic to right coset representatives obtained from specific subgroups of the associated affine Weyl group.