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Abstract Kontsevich and Soibelman defined DonaldsonThomas invariants of a 3d CalabiYau category equipped with a stability condition [41]. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Given a stability condition, the DT-transformation allows to recover the DT-invariants. Let S be an oriented surface with punctures and a finite number of special points on the boundary, considered modulo isotopy. It gives rise to a moduli space X P G L m, S, closely related to the moduli space of P G L m-local systems on S, which carries a canonical cluster Poisson variety structure [13]. For each puncture of S, there is a birational Weyl group action on the space X P G L m, S. We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution of X P G L m, S

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