  We prove homological mirror symmetry for the universal centralizer $J_G$
(a.k.a the Toda space), associated to any complex reductive 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 the center of $G$ is
not connected).
