Is it permissible for the manager in charge to allow his team to leave for their homes before the end of official working hours if there is no work or complaints, knowing that his director has authorized him to do so?

There is no objection to your granting permission for employees to leave after work hours and before the end of official duty time, so long as your superior has authorized you to do so. You may grant permission to those who need or deserve it as an incentive, or you may retain those you might need in case of emergencies. There is no objection to granting permission to all of them if your manager is authorized to do so.