"Edmund Burke argued that the only thing a representative owed his constituents was his own best judgment."
To which I would ad, he owes a clear honest representation of his position on salient issues of the day, eg, a governor on capital punishment. If he supports it, he owes it to his state to say so. Then he owes it to them to govern accordingly, if elected.
But if a candidate wins, is it up to him to act according to the dictates of his conscience, rather than the desires of the electorate? Politicians can always fudge or reconcile these things. They're generally elected on issues like the economy and competent management of the government, but if they get in, they freely interpret their election as a mandate to act as they see fit on other issues. Is that legitimate?
I don't know what the answer is. One anti-death penalty governor didn't commute every sentence, because he assumed he didn't have the authority, and he made a good case for his view in print (unfortunately for him the public resented the cases where he did commute sentences and defeated him at the polls -- I think it was DiSalle of Ohio in the 1960s). The counterargument is that if the criminals are still behind bars no harm has been done, but I don't know if that's valid. If the death penalty has a deterrent effect and voters want capital punishment for that reason, isn't the governor violating their trust?
Perhaps an anti-death penalty governor could make a good case for following his conscience, but plenty of politicians have claimed to be voting the demands of their conscience when outsiders saw little more than politics and ambition behind his actions. State reps who make the decisive votes against the death penalty are often rewarded for changing their minds, and there's been much uproar in Illinois about the previous Governor's use of the pardoning power, and what may have been behind it.