My future husband showed me a court letter of polygyny permission. How do I know it is genuine?

Check with the authorities; do not rely on how the letter looks alone. A court order has details such as a case number, the name of the court, the date and the names of the parties. Note those details and ask the Islamic religious affairs office that will process your marriage, because they must check the order before granting permission to marry. Also see whether the prospective wife's details in the order match yours. If he is reluctant to let you or your family check, that is a serious warning sign. This check is not excessive suspicion, but a duty to yourself and your future children. See checking marriage documents.