Below my right side passenger peg on my 2001 FXDWG is this filter. I'm thinking its a breather of some sort. The hose it connects to runs up under the tank to either the rocker boxes, carb, or gas tank. I can trace it past the rear rocker, but lose it there, as I haven't removed the tank.
So the filter unit broke off. It seems there should be a flange the hose attaches to the has come off. Part of this broken flange is still in the hose, held in with a hose clamp.
So, what's it called? I need to get a replacement. It's pretty oily, that's why a breather came to mind....

 

Ok multiple pics didn't work. Here's the actual filter, the previous pic was its mounting point and connecting hose.

 

Could it be off the tank vent or crankcase breathers?

 

looks like its a breather filter to me...DK customs sells those. He is a forum sponsor...find his name to the right

 

Here is how the breather is routed on my Softail, those are the crankcase and transmission breather hoses. A filter on the end serves no purpose other than to get nasty looking.