What is the ruling on the portion of the salary related to the hour that an employee does not work, when his manager has permitted a reduction in working hours?

If there is no work for you and your manager permits you to leave, then there is no harm in it for you. The manager, however, must fear Allah and fulfill his trust. But if the work requires your presence, then it is not permissible for you to leave, even if the manager gives you permission, because the work might need you. This is what the two Sheikhs, Ibn Baz and Ibn Uthaymeen, clarify.