  Continuing the study of perverse sheaves on the nilpotent cone of a
$\mathbb{Z}/m$-graded Lie algebra initiated by Lusztig--Yun, we study in this
work the parabolic induction and introduce the notion of supercuspidal sheaves
on the nilpotent cone. One of our main results shows that simple perverse
sheaves with nilpotent singular support (called bi-orbital sheaves) are
produced by parabolic induction from supercuspidal sheaves. As application, we
provide a proof for a theorem announced by I. Grojnowski on the parametrisation
of simple modules of affine Hecke algebras at roots of unity via
Deligne--Langlands--Lusztig parameters with nilpotent singular support.
