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?