A thrust reverser for an aircraft propulsion assembly, including a fixed structure equipped with at least one cascade as well as a radially inner delimiting wall of a secondary vein, and a movable structure including at least one reverser cowling, the cascade being arranged, in the forward direct thrust position, in a housing of the reverser cowling, and, in the retracted thrust reversal position, the wall reveals a passage opening of the secondary vein towards the cascade, the thrust reverser also including at least one sealing membrane designed to deflect at least one portion of the secondary flow. A hook is provided for securing the membrane to a rear cascade support frame, and to the radially inner delimiting wall of the secondary vein.