  This paper gives a combinatorial description of the set of irreducible
components of the semistable locus of the global nilpotent cone, in genus
$\ge2$. The first main result of this paper states that the set of irreducible
components of the global nilpotent cone is given by the very natural
decomposition in twisted Jordan strata, which are smooth. Then we move on to
the semistable locus and obtain purely combinatorial conditions on the twisted
Jordan type to be semistable. The proof uses an analogous result obtained in
the context of moduli stacks of chains, and shows that semistability can be
tested on the most `simple' subsheaves - the ones built with iterated kernels
and images. The proof is constructive and do not rely on the coprimality of the
rank and degree, in particular we get that the attracting cells are irreducible
in any case. The last section describes this set of semistable components in
terms of integral polytopes.
