When analyzing the behavior of finite-state concurrent systems by model checking, one way of fighting state space explosion is to reduce the model as much as possible whilst preserving the properties under verification. We consider the framework of action-based systems, whose behaviors can be represented by labeled transition systems (LTSs), and whose temporal properties of interest can be formulated in modal mu-calculus (Lmu). First, we determine, for any Lmu formula, the maximal set of actions that can be hidden in the LTS without changing the interpretation of the formula. Then, we define dsbrLmu, a fragment of Lmu which is adequate w.r.t. divergence-sensitive branching bisimilarity. This enables us to apply the maximal hiding and to reduce the LTS on-the-fly using divergence-sensitive tau-confluence during the verification of any dsbrLmu formula. The experiments that we performed on various examples of communication protocols and distributed systems show that this reduction approach can significantly improve the performance of on-the-fly verification.