Rory speaks of Luke and says to Lorelai "You called him Duke for two years just to make him mad."
Originally, the diner owner was supposed to be a woman named Daisy, but was then changed to a male character. According to Amy Sherman-Palladino, she just changed the name and didn't make any changes to the dialogues. In another interview, Scott Patterson said that the second dialogue he got for the audition had his character misspelled as Duke (instead of Luke). Rory's line seems to be an inside joke about that.
When Lorelai and Luke are pretending to be a couple to view the apartment, Lorelai says she could wear nothing but cellophane and a Yankees cap and Luke wouldn't pay attention to her. In real life, actor Scott Patterson, who plays Luke was a minor league baseball player and played in the Yankees organization.
The book Kirk is trying to buy cheaper in the book fair is Like Water for Chocolate, a Mexican novel by Laura Esquivel which was also adapted into a movie in the '90s.
After being located just beyond the kitchen area for the first season on, the bathroom in Luke's place has moved to the other side of the apartment near the bedroom area. It remains there for the remainder of the series.