Well the Falchion is pretty much in the Auctioneer's hand, it's pretty obvious that the weapon was a slaver's, and is thus evidence. As such, it is likely that it will be 'bagged and tagged' as it were, stored away to eithe be reclaimated as part of the Armory's wares, or perhaps put on sale to further the growing economy after such financial struggles as war often brings.
If you move to claim it, the Captain and Paladin would most likely question you as to the validity of your claim to it, boring you with legal jargon and what not before the Paladin walks off for a spell.
If you're insistant on owning it, the Captain will mention stopping by the Garrison the following day, and that perhaps a financial contribution to the efforts of quelling future slave auctions may enable him to lose it in the paperwork, but to only bother if you've a serious ofer to back your interest. The he will return to work.
Oh, and leaving the black curtained room, you will find yourself in a cellar beneath a forclosed building scheduled for renovations, about a twenty minute walk from the Green Gryphon.