Obviously in reference to the Trump immunity case, but I’m not looking to discuss that. As I understand it, it takes 4 of the 9 justices to decide to hear a case, but after that, who decides when oral arguments occur? Ref to rule/statute/whatever would be appreciated.
Generally speaking, under Supreme Court Rule 27, the Clerk of the Court is responsible for maintaining the calendar, including setting cases for argument.
The other rules generally govern the other aspects of the schedule – when briefs are due; when the matter is distributed for consideration; and so forth.
For deviations from this, either by motion or sua sponte, it’s going to be by order of the Court. So, if you’re asking who decided that the immunity cases is going to be heard on an expedited schedule, then the answer is the Court.
If you’re asking how many justices are required to order that, I assume it’s a majority, but I don’t know for sure.