According to the Targum Sheini’s interpretive translation, Haman’s oldest son, Shimshi, cast the pur.
The Malbim writes that the lot was cast for Haman – it decided when he would die since this plan to kill the Jews led to his execution (see below Esther 7:10).
The Chassam Sofer and the Me’am Loez write that Haman saw himself on top, and the Jews beneath him. Unfortunately for Haman, he did not interpret this correctly, as it was pointing to his hanging from the tree, and Mordechai beneath him, standing safely on the ground.
The Maharal says that Haman did not throw the lots himself for two reasons. The first is that he knew he was not a spiritually sensitive person. He therefore asked someone else to interpret the lots. The second reason is that he knew he had a subjective bias in the result. As such, his own subjectivity would subconsciously color his interpretation of the final result of the lots tossed. Perhaps these two answers are really one and the same. One cannot be a spiritually attuned person with biases. The more spiritual one becomes, the more objective one becomes. The entire goal of spirituality is to realize that our own wants should not have significance.
The Ben Ish Chai points out that the way Jews escape annihilation is through their performance of mitzvos, of which Torah study is the greatest (see Mishnah, Peah 1:1). That being the case, the Ben Ish Chai interprets our verse in a novel manner by writing that the lots cast pointed to the solution to Haman’s challenge being “lifnei Haman” the letters preceding the letters in Haman’s name in the Hebrew alphabet. The letters before hey (dales), mem (lamed), and nun (mem) can form the word “lamed,” (“learn”).
The Sfas Emes writes that Haman felt he needed to pick the right day of the week, as well as the correct month. Since, the days of the week represent the natural cycle established in the seven days of Creation, and every culture has its own fashion for establishing and measuring months, days represent a variable given Divinely. Months, however, represent a variable provided by people. Haman therefore thought the rabbis, who established the Jewish months (as mentioned above), were prone to error. Hence, Haman felt he could not be successful against H-Shem, Who established the days, but could be successful against the rabbis, who in his view represented imperfect, fallible men.