The Yad HaMelech points out that Achashverosh stresses that Mordechai sits at the king’s gate in order to allay Haman’s concern that honoring Mordechai may be beneath his dignity. This is not to imply that Achashverosh suddenly cares about Haman’s honor; in fact, if Haman were to honor somebody lowly, that could reflect poorly on Achashverosh, his king.
According to the Talmud (Megillah 16a), Haman attempted to stall by pretending to not know which Mordechai Achashverosh had in mind, and the king had to narrow down the identity of this particular Mordechai.
The Maharal says that the reason Haman’s advice of giving the honoree the crown (Esther 6:8) is not mentioned again is because the king should ordinarily give these items, himself. However, in the case of this person who is a major adviser who “sits at the gate of the king,” Haman can bring him the crown.
The Ginzei HaMelech notes that Achashverosh here does not describe Mordechai as one “who sits at my gate,” but rather the “gate of the king” because he is alluding to the fact that Mordechai sits at the gate of the King, H-Shem. Earlier (Esther 2:5), when Mordechai was referred to as the “son of Kish,” the Talmud (Megillah 12b-13a) understood that to mean that he knocked (“hikish”) at the Gates of Mercy. His praying therefore qualified him to be called one who sits at the gate of the King.