Why did Esther request that Haman's sons be hanged when they were already dead?
Haman, because he felt he was insulted by Mordecai, decided to gain revenge by destroying not just Mordecai but all the Jews. This was because Mordecai would not bow to Haman as he passed in the streets. Haman was told that Mordecai refused because he was a Jew; thus, Haman decided to wipe out all Jews.
He implemented the plot subtly by having a law passed to wipe out the Jews. "Then Haman said to King Ahasuerus, "There is a certain people scattered and dispersed among the people in all the provinces of your kingdom; their laws are different from all other people's, and they do not keep the king's laws. Therefore it is not fitting for the king to let them remain. If it pleases the king, let a decree be written that they be destroyed, and I will pay ten thousand talents of silver into the hands of those who do the work, to bring it into the king's treasuries"" (Esther 3:8-9).
When Esther exposed Haman, the problem was that the Medes believed that a law could not be revoked. The intention was to make sure proper consideration was given before invoking a law, but when someone became careless, this could lead to severe problems. The solution was to amend the law to allow the Jews to defend themselves if they were attacked. "By these letters the king permitted the Jews who were in every city to gather together and protect their lives-to destroy, kill, and annihilate all the forces of any people or province that would assault them, both little children and women, and to plunder their possessions" (Esther 8:11).
For the most part, this took the spirit out of the opposition (Esther 8:17). But some still tried to kill off the Jews in order to gain their property (Esther 9:1). Among those killed who were trying to kill the Jews were Haman's ten sons (Esther 9:10). They did not heed what had happened to their own father and tried to take revenge.
Hanging was considered the most shameful mode of death. Though these men were already dead, by hanging them instead of giving them a respectful burial, they were exposed to the ridicule of everyone. Remember that there was still an element in the kingdom who wanted the Jews dead. By this public destain for the anti-Jewish element, the last of the opposition would be squashed. It is an interesting solution because it stopped the violence without harming any additional people.