  We prove homological mirror symmetry for the universal centralizer $J_G$
(a.k.a Toda space), associated to any complex semisimple Lie group $G$. The
A-side is a partially wrapped Fukaya category on $J_G$, and the B-side is the
category of coherent sheaves on the categorical quotient of a dual maximal
torus by the Weyl group action (with some modification if $G$ has a nontrivial
center). This is the first and the main part of a two-part series, dealing with
$G$ of adjoint type. The general case will be proved in the forthcoming second
part [Jin2].
