I visited the campus for an in-person interview 2 weeks ago. They supported my flight and hotel. I thought I was good at the interview and the director stated that she will be in touch after several weeks. Is it okay if I send follow-up email? I am so nervous and cannot focus on my work. Any comments will be appreciated.
Unless they gave you a specific time frame, then yes, it is fine to ask them about the current state of your application.
If you have other options, then keep them open until things are definite, but don't bring them up when asking for an update.
Don't become a pest, but a couple of weeks is a pretty good interval.
I would say that if no time frame is specified, two weeks is enough to warrant an email. I've actually had cases where I had been accepted to a position, they just hadn't informed me yet. It is likely that they may still be interviewing other candidates or have not made a decision so it will do you well to keep your email straight and to the point, IE
Thank you for taking time to interview with me, I was wondering if there were any updates on the position.
You could maybe add a thank you to their hospitality but I find most people in academia don't like emails, so besides adding the standard niceties, I try to get in and out quick.
Note: my field is CS, so small talk is usually at a minimum. Some people find this type of communication a little too impersonal.