Getting a letter of recommendation from a socially controversial PhD advisor I am about to finish my PhD at a US university and am starting to apply for jobs and postdocs in physics and computer science. Throughout my PhD, I maintained a good relationship with my PhD supervisor. However, in the last year and a half, he has made social (not academic) comments that were very inappropriate. These comments more directly influenced other students, and the proper actions have been taken with the university and an investigation is on-going. To be clear, the comments are inexcusable and I have lost all respect for my supervisor.
I am wondering what the most appropriate way for me to proceed is, as many positions explicitly ask for a letter of recommendation from ones supervisor.
Some additional data:

I, luckily, had a rather successful PhD and have many collaborators who I could ask in his place. However, I imagine I would need to address why my advisor is not writing a letter.

I do think my advisor would write me a positive letter, but (a) I feel uncomfortable asking him for a favor and (b) I worry that (and I hope) if public action is taken, his letter will not be viewed in a positive light. Quite simply: I'd rather not have a word of support from him.


Is this something I can/should address in a cover letter or otherwise?
While there is always risk in such things, if his professional credentials are good and the other comments don't reflect badly on that, I'd suggest asking for the recommendation and not otherwise mentioning it in cover letters and such. Treat it as a purely professional relationship, which it should be.
As you note it is odd to not have a recommendation from one's advisor. And I'm assuming that your judgement is that his letter would be professional. If you have a way to find out anything about what he has written for others, it would be a benefit. A department head might, possibly, give you some guidance. But that is another judgement call - whether to ask.
Of course, you can't control the reactions of others, but you can, if asked, disassociate yourself from his views. Just say that you don't hold those views in any interview or such.
I'll note that in math and cs there have been a number of similar situations where a person is widely respected professionally, but whose views on some issues have been shunned and the people condemned. It doesn't reflect on their technical work, not should it reflect badly on you or your work.

See the case of William Shockley, and that of Robert Lee Moore.
And, of course, many people hold appalling views of things, that may be known to others, but not to their students.