Gi
ven a curve X of genus g\, the moduli stack of Higgs sheaves of rank r and
degree d is known to be of dimension 2(g-1)r^^{2}. It can be view
ed as the cotangent stack of the stack of coherent sheaves of type (r\,d)
over X\, and Laumon proved that the substack of nilpotent Higgs pairs is L
agrangian. This substack is a global analog of the nilpotent cone\, and is
nothing but the 0-fiber of the Hitchin map. It is highly singular\, and o
ne first interesting step toward its comprehension is the study of its irr
educible components. This study is also motivated by a result stating that
the number of stable components is given by the value at 1 of the Kac pol
ynomial of the quiver with one vertex and g loops (conjectured by Hausel\,
Letellier\, Rodriguez Villegas\, proved by Mellit)\, as well as by the W=
P conjecture (de Cataldo\, Hausel\, Migliorini). I will give a nice combin
atorial description of this set of components\, and will explain which one
s subsist when we restrict ourselves to the semistable locus (with respect
to the usual slope stability).