What does Colonel Sartoris do for Miss Emily after her father's death? Why?
After Emily's father dies, Colonel Sartoris invents a way to help Emily survive financially without embarrassing her by revealing that she's actually accepting charity from the town. Everyone has to pay property taxes on their houses, of course. But Colonel Sartoris permanently excuses Emily from paying those taxes, telling her a lie about how her father had given money to the town, and now they are repaying him by cancelling all of her taxes. Again, he does this for her as a favor, saving her from poverty, keeping her in her own house, and preventing the embarrassment she would feel if she were offered a more straightforward method of charity.
Here's the explanation from the narrator, which we find in the third paragraph of the story:
"Alive, Miss Emily had been a tradition, a duty, and a care; a sort of hereditary obligation upon the town, dating from that day in 1894 when Colonel Sartoris, the mayor—he who fathered the edict that no Negro woman should appear on the streets without an apron—remitted her taxes, the dispensation dating from the death of her father on into perpetuity. Not that Miss Emily would have accepted charity. Colonel Sartoris invented an involved tale to the effect that Miss Emily's father had loaned money to the town, which the town, as a matter of business, preferred this way of repaying. Only a man of Colonel Sartoris' generation and thought could have invented it, and only a woman could have believed it."
This invention of Colonel Sartoris is important to the plot of the story, but it's also thematically significant. Emily clings to the inaccurate, bygone assurances of Colonel Sartoris even after he's dead—she tells the new government representatives that she has no taxes in Jefferson, and she refuses to pay. She even tells them to take up the matter with Sartoris himself; it's as if she doesn't understand that he's dead. Her tenacious insistence on this false notion of having no tax obligations is not just self-deception on Emily's part but is also an outward sign of how she is holding onto the past.