I like Sherman Alexie and this book is semi-autobiographical. There is a lot about identity and racism. It tends to appeal more to boys. If I had to compare his experience to something, I would say that he is like a Boston kid coming to New England suburbs through the METCO program, since he chooses to go to a school off the reservation (obviously this contributed to Alexie's access and ability to move within the "white man's world" in that he is a successful, published author and there are almost no other Native American authors as successful in the U.S., except maybe Joseph Bruchac.)
Having said that, the biggest controversy is that the book from the Native American perspective is that it's about reservation life. I know, sounds weird when you are looking for a book about the authentic Native American experience. I will repeat, it's semi-autobiographical, so it is "authentic" to the author's experience. However, only 25% of all Native Americans live on the reservation, so keep that in mind.
In addition, because it is reservation life, there is the problem with "reinforcement of stereotypes" because of the poverty and alcoholism that he depicts. Again, this was his experience, and alcoholism is indeed an issue among Native People, but it is not the only experience of being Native American.
The book did not "stick with me" - hence I read it twice. I prefer the books by Thomas King for multiple reasons, but they are not novels. They are short stories, written the oral tradition style, and each one can be used to tackle an important historical or cultural issue.
Other controversy comes from, as you could guess, white families: https://www.goodreads.com/