In "The Great Gatsby", why do Tom and the sloanes snub Gatsby after asking him to dinner?
The incident occurs when Tom, Sloane, and a young woman were riding horses one afternoon and they stop by Gatsby's house for some refreshment. They stop mostly because they wanted a rest and something to drink, but also probably because Tom was curious about Gatsby and how and where he lived. It's clear from the first comment Sloane makes that Sloane sees himself as superior to Gatsby. He is rude in his abrupt comments and in his lack of desire to join in the conversation. The young woman, after two drinks, opens up some and is more friendly. She asks Gatsby and Nick to join them. Sloane does not want any part of that arrangement and whispers something to her. When Gatsby leaves the room to get ready, the three have left. Nick understands from the moment they arrived that they are not there because they like Gatsby. They are simply using him as a pit stop and because Tom wanted to know more about Gatsby since he is suspicious of Daisy's activities and involvement with Gatsby. The three snub Jay Gatsby because they see him as a nouveau-riche upstart who is beneath them in every sense. He doesn't have the disinterested demeanor that they, and their crowd, possess. He is too eager to please because he doesn't have the long-standing past of money and all that it brings. They, along with Daisy, Jordan and others like them, are the careless people that Fitzgerald talks about in the last chapter.
This question needs rephrasing before it can be answered completely; only the young woman asks Gatsby to dinner. Mr. Sloane and Tom did not want Gatsby to join them for dinner, which explains the "impassioned conversation" between Sloane and the young woman on the porch.
The reasons why the young lady wants Gatsby to join them for dinner are not completely clear; before inviting Gatsby to dinner, she has invited herself to Gatsby's next party, so perhaps she wants to make a social connection to Gatsby, or she is suddenly feeling competitive with Gatsby and wants to assert herself as a fellow party-thrower. No matter the reason she wants Gatsby to come to dinner, Tom and Mr. Sloane do not want Gatsby to join them. Tom is already feeling threatened and exasperated by Daisy's attachment to Gatsby, and these negative feelings surface when Gatsby makes it clear that he would like to come to dinner.