How can I install an air conditioner in a window with a very deep exterior ledge?
I strongly recommend asking your landlord. They will have dealt with this before, and will know what solutions will work -- and given the potential liability issues they may insist on doing the instillation.

(I have dismounted a NYC airconditioner, which had been installed with a moderately complex metal framework apparently designed to solve exactly the problem you are describing. I don't remember any details, though.)