I will present work in progress on the homological mirror
symmetry for the universal centralizer $J_G$ associated to a
semisimple Lie group $G$. The A-side will be a partially wrapped
Fukaya category of $J_G$, and the B-side will be the derived category
of coherent sheaves on the GIT-quotient of the dual torus of the
maximal torus by the Weyl group action (with some modification if $G$
is not of adjoint type).