Assuming here the USA (because it's mostly the USA that does this kind of rabid shit): most states have "right of publicity" laws in which individual have exclusive right to control the commercial use of their name, likeness, or other recognizable aspects of their persona. Commercial use of someone's such identity and/or persona can be very, very severe depending on which state the offending action was taken in.

More federally we have the Lanham act that explicitly prohibits the use of a person's name/identity in commerce in a way that is likely to cause confusion about affiliation, sponsorship, or endorsement. This doesn't even require a trademark registration, though it is easier to get your Lanham rights if you have one.

Finally there's the hallucination angle which can lead to defamation suits. The LLM will inevitably hallucinate, and if the hallucination is of the variety that could expose the named person to ridicule, hatred, etc. that is a straightforward defamation suit that could result in a whole lot of monetary loss to the publisher.

And here's the fun part. These are not exclusive. A publisher could face statutory damages for right of publicity violation, stripping of profits for the Lanham violations, and compensatory and punitive damages for defamation by hallucination.

Note: IANAL, obviously, but I do research things for a living. Consult with an attorney, not me, for actual court cases.