Capital City: Of the Associated Worlds as a whole, that would be the Conclave Drift, the giant habitat in which the Conclave of Galactic Polities is situated. It’s also a major commercial and cultural hub, it being – due to the room it sets aside for every polity in the Accord to build its own little mini-city – one of the few places you can find everyone together, and its markets are one of the few places in the Worlds where you can rely on finding just about everything that it’s legal to buy anywhere. (The Empire had this in mind, of course, when they donated one of their star systems to build the thing in.) It is slightly subverted inasmuch as not all that much freight gets transshipped through it; it would be rather out of the way. But a surprisingly high percentage of the actual deals get made here.
Other candidates for major commercial centers would be Mer Covales, on Seranth (Imperial Core), which houses several major commercial exchanges and which does play host to a great deal of manufacturing and transshipping; and the worlds of the Free Eilish Confederacy, whose policies of neutrality and openness make it a favorite spot for business – and also a favorite spot for back-door politics and for galactic intelligence agencies to host their away games. Neither of those, of course, are political capitals of anything.
Of the Empire, that would be Calmirie (“center of order”), which is both the political capital of the Empire, and a significant commercial (somewhat overtaken by Mer Covales) and cultural (somewhat overtaken by Delphys (Imperial Core)) center. It plays it essentially straight.