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I will present recent results about Homological mirror symmetry (HMS) of algebraic symplectic varieties coming from representation theory. The first is about HMS of the universal centralizer of a complex reductive group G. The second, which is a joint work with Yun, is about HMS of a moduli space of Higgs bundles on P^1 with certain automorphic data. The latter variety is closely related to the first one as it is a symplectic compactification of the former. No prior knowledge about HMS is required.