Board Thread:Skyrim/@comment-101.162.73.24-20150706040326/@comment-70.170.55.96-20150706061420

Maybe because she's not in the Guard Faction. Followers and NPCs that are in that faction will attack you if you get a hefty bounty in their main Hold location. If I recall correctly, I believe Delphine is in that faction and she will turn hostile if your bounty is high.