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Haman the Viceroy: 
After the above events Haman was appointed by the king as the viceroy. All subjects of the king were required to bow to Haman upon his arrival. Mordechai however refused to bow.
Who was Haman?
Haman was a descendent of Agag, the king of Amaleik. Haman’s grandfather was born as a result of the delayed execution of Agag. This delay was caused by Shaul the ancestor of Mordechai. Some say that in fact Haman was not at all a descendant of Amalek but was figuratively listed as his descendant to his great wickedness.
The lawyer: Haman had a rich record of anti-Semitic activity. He acted as the legal representative of the Samaritans to fight the Jews from building the Temple during the reign of Koreish.
Slave: Haman was actually the legal slave of Mordechai. The story behind Haman’s acquisition by Mordechai as a slave, is connected with a certain trip Haman made together with Mordechai and others. When Haman ran out of food he needed to sell himself as a slave to Mordechai in exchange for the food.
Advisor: Eventually Haman was appointed as an advisor of Achashveirosh. His original name was Memuchan, and it was he that advised the king what to do with the insubordinate Vashti.
Why was Haman given such a high position by Achashveirosh?
This is because Haman was very wealthy. Rebbe Pinchas stated that there were two very wealthy people in the world, one from the Jews and one from the gentiles. Korach was the wealthiest Jew as he found the treasures hidden by Yosef. Haman was the richest gentile in the world as he stole the treasures of the Beis Hamikdash. Achashveirosh saw his vast wealth and hence desired to raise him into power. Alternatively Achashveirosh was incensed that Mordechai wanted to rebuild the Beis Hamikdash and he thus elevated Haman, the Jew hater, to power in order to prevent Mordechai from following through with his desires. Alternatively Achashveirosh mistakenly thought that Haman was the one who saved his life against the assassination attempt, as the scribes of the king were the sons of Haman and they lied and wrote that it was Haman who saved the kings life. Thus as reward, Haman was appointed viceroy.
Why did Hashem allow Haman to be given such a high position?
Hashem wanted the entire world to recognize His greatness and the greatness of the Jews. Therefore He elevated him to the point that he would be well known to all, and then had him executed on the pretense of his hatred to the Jews.
Why did Mordechai refuse to bow to Haman?
Haman wore an idol on his clothing and desired for the Jews to bow to idolatry. Alternatively [Haman did not wear an idol but] Mordechai did not desire to flatter Haman by prostrating himself to him. Upon being asked why he refuses to bow, as even the forefathers bowed to Eisav, Mordechai replied that he comes from Binyamin and Binyamin was still a fetus at that time and hence Binyamin did not bow, so too he will not bow.
 Megillah 13a; Yalkut Shimoni Beshalach 268; Tana Dvei Eliyahu Raba 24; Likkutei Sichos 3/913
 Yerushalmi Yevamos 2/6 “Was Haman in truth the son of Hamdasa? No. He was a Rasha the son of a Rasha”; See Likkutei Sichos 14/90 that brings this as one of the suggested explanations of how it was permitted for Esther to keep the house of Haman, as in truth Haman was not from Amalek. The Rebbe negates this conclusion.
 See Yalkut Shimoni 1045
 Megillah 15a Rashi “Vizeh Ba”
 Esther Raba 11; Agadas Esther 3/1
 Yalkut Shimoni 1053
 Iyun Yaakov Megillah 13b
 Yalkut Shimoni 1053; Esther Raba 11
 Esther Raba 11; Yalkut Shimoni 1054
 Yalkut Shimoni ibid