This topic was discussed in this thread (http://forums.civfanatics.com/showthread.php?s=&threadid=22492).

I am pretty neutral on this issue but I'd like to give my spin on it:

If being the Governor of the Capital Province is a requirement of successfully running the Domestic Department then the Domestic Deputy should also be the Deputy Governor.

If being the Governor is an ancillary duty or a perk of the office then the Capital Province should be run the same way as every other province. That is, no Deputy Governor.

EDIT: I have decided to abstain. I can fully agree with either of these options and fully support the winner in the game. Any selection I made would be random.

since all other provinces have no deputy, i say NO

I say yes, because the Domestic leader is governor of our first province because it is generally the most important/productive and needs more leadership. Also, the deputy could work out some of the details for the leader, since he has so many other responsibilities.