  In this paper, we prove centralizer rigidity near an element of the Weyl
chamber flow on a semisimple Lie group. We show that a volume preserving
perturbation of an element of the Weyl chamber flow on a quotient $G/\Gamma$ of
an $\mathbb{R}$-split, simple Lie group $G$ either has centralizer of dimension
$0$ or $1$, or is smoothly conjugate to an element of the Weyl chamber flow. We
also acquire a general condition for the centralizer of a partially hyperbolic
diffeomorphism to be a Lie group.
