Because my ex-wife has allowed her boyfriend to move in with her, can I now reduce or stop paying alimony entirely?

Your former spouse being in a supportive relationship may be grounds for a modification of alimony. This can only happen if the court ordered the alimony in the first place or if you both reached a marital settlement agreement that left the alimony as modifiable. You should consult with an attorney on this issue to ensure that you qualify for a modification.