Interactions between autoequivalences, stability conditions, and moduli problems

We formulate a strong compatibility between autoequivalences and Bridgeland stability conditions and derive a sufficiency criterion. We apply this criterion to an extension of classical slope on the derived category associated to any Galois cover of the nodal cubic. In particular, we give an explicit description of the moduli space of stable objects, its compactification (under S-equivalence), and the group of all autoequivalences compatible with the choice of stability condition.