If the head of the jazz program recommends you to admissions and you get in, I expect he’d be a bit disappointed if you didn’t participate at all in jazz at Columbia. But there’s certainly no obligation and I don’t expect anyone would try to pressure you. The jazz folks at Columbia are a pretty laid-back group, including the head of the program. In addition, what would be the point? You don’t want to be playing with someone who doesn’t want to be there.
I should also say that there’s a pretty wide range of opportunities to do jazz at Columbia, some more intensive and some less. When I refer to a “jazz program”, that’s really shorthand for the whole array of opportunities, and each student can decide for himself or herself how much of it they want to do.