What Emily inherits from her father that most affects her adult life is her attitude about her position in society. As a Grierson, her father was considered to be of the elite class in Jefferson, and as such, her father didn't think that any of the young men where "quite good enough for Miss Emily." The townspeople pictured Emily and her father as a portrait with her father in completely dominant position at the forefront of the picture, and Emily, barely visible behind him. The superior attitude and the subsequent behavior leave Emily alone after her father's death, and she is more desparate than ever to keep Homer, so she takes the ultimate control and kills him and then keeps his body in the upstairs bed chamber.
Ironically, the only thing she actually inherits is the house they lived in and little to no money, so she is actually not the most wealthy or elite person in town, and is now at the mercy of the town elders, who for the most part treat her with the respect her father deserved, but that too isolates her and makes her take her desparate actions.