A Cordial Invitation will glitch if Bekarna's Observatory is cleared out and the puzzles solved before completing The Archmage's Vault quest.
Once Bekarna's Research is collected, invading the manor through the front door and entering the study will trigger the imp to inform the player that Arkemyr is waiting in the Scrying Pool room before becoming hostile. Arkemyr can be found in the Scrying Pool room, and will become aggressive immediately.
After The Archmage's Vault is completed and A Cordial Invitation quest is received (with Bekarna's Notes), Arkemyr does not appear in the study to progress the quest. He is in the Scrying Pool, and interaction with him results in no dialogue. When interacting with the pool, the Circle of Magi will interact with you as if you have already progressed the quest, and will take the research. The quest does not complete.