Well, that clears things up a lot. That body you are looking at is called the Auxiliary Pump. On other HSTs, it's called a charge pump. It functions to supply the main HST with fluid to prevent cavitation. The big hex nut is the cover for the aux pump's relief valve. The Maint. Man'l says this pump puts out 498 psi and a little less than 1-1/2 gpm flow at 1800 rpm. It also supplies the hydraulic lift cylinder for lifting the deck.
The maint. man'l says nothing about this being a PTO stub, but there wouldn't be much reason for it to be there if it didn't power something. It sounds like your dealer knows exactly what it is use for. I'd bet this shaft is called out and described only in the manuals for the add-on accessories.
Thanks for the photo. That made it much easier.