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Do you have an employment contract stating that you can only be terminated for cause?
Ok. Two issues here.
The fact that you may be having an extra martial affair does not really fit the definition of "hostile work environment." The Supreme Court gives us the definition of that, which requires severe harassment based on a person's race, religion, gender, age, disability or FMLA use. So, if you were having an affair AND demanding that all women in the office also engage in sexual relations with you, then we'd be talking about a hostile work environment.
That being said, without an employment contract you are an "at will" employee. That means that, legally speaking, they don't need a good reason to fire you or request your resignation. So, to address just your question "can they ask me to resign?" yes they can, but it really has nothing to do with whether or not you may be having an extra-marital affair. It has more to do with that fact that they can ask you to resign anytime they want.
All that said, don't resign. Make them fire you so that you get unemployment. An extra-marital affair outside of the office is not considered misconduct sufficient to block your unemployment.