  In this paper we study the Fourier transform on graded Lie algebras. 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
$\chi(\mathbb{C}^\times)$. Here under some assumptions on the field $\Bbbk$ and
also assuming two conjectures for the group $G$, we prove that the Fourier
transform sends parity complexes to parity complexes. Primitive pairs have
played an important role in Lusztig's paper \cite{Lu} to prove a block
decomposition in the graded setting. A long term goal of this project is to
prove a similar block decomposition in positive characteristic. In this paper
we have tried to understand the primitive pair and its relation with the
Fourier transform.
