Short-listed, but no interview invitation after three weeks - should I follow up?
Congratulations on making the short list! There is no way to tell from the outside what the school is doing, but there is also little that you can do to change it at this point.

If the school's answer would change something for you in the short term - for example, if you have another job offer and want to know if the school is still considering you - then it would be very appropriate to send them an email.

If you are just emailing them to check on the status of your application, I would wait a few more weeks to see if you hear anything from them. This part of the application process is often a waiting game, unfortunately.  On the other hand, I think that a very polite, short message would not be likely to harm your application, particularly if you let them know that you are still very interested in their department.