  Let $G$ be a complex, connected, reductive, algebraic group, and
$\chi:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on
$\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of
$\chi(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity
sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and
the group $G$. The assumption on $G$ holds for $GL_n$ and for any $G$, it
recovers results of Lusztig in characteristic $0$. The main result is that
every parity sheaf occurs as a direct summand of the parabolic induction of
some cuspidal pair.
