1) You're upfront about it and tell your manager, they update workday, HR schedules a meeting with you to explain the multiplier effect and how your salary is being reduced.
2) You're kind of sneaky, you update your state tax withholding, it updates workday and put you into the appropriate COL bucket for next year's performance review, your salary slowly falls in line with people in your COL area.
3) You keep it secret, allow the company to keep paying taxes to your original state, maybe pay estimated taxes or something to your new state? Don't develop any relationships with coworkers that'll eventually leak your true situation, get away with it?
But if you reduce my pay, you're gonna make me take stock of what I'm giving you and shift the balance in my favor if I can easily get away with it.
EDIT: And odds are, the employer won't notice enough to fire me.