I will present work in progress on the homological mirror symmetry for the universal centralizer JG associated to a semisimple Lie group G. The A-side will be a partially wrapped Fukaya category of JG, 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).