Notifying the courts or not

Yes, she is your ex, you have nothing to do with what she is doing.
Why do you know ? Why do you care ?

Perhaps there is a lot more here and you are not properly asking the question ?

Is there child custody or children involved ?
But even so, unless the child custody agreement says something about it, she is allowed to move a new boyfriend in. he does not have to tell you or the court.

** this is based on general US laws, your location could be different