Is your bike a california model?

If so, it appears to be the PAIR control valve. Im not 100 percent sure what it does, but is a pulse secondary air injection control valve. Its a smog thing. Again, Im just taking a stab at it as Im not 100 percent sure.

If gas is coming out of it, I would assume you have one or more floats stuck in one of your carbs.

Oh, and the hose doesn't go anywhere, its designed as a vented overflow and it dumps the excess gas out the line. the line is tucked away behind the water pump and out the bottom so the gas gets away from the bike. It should also be in a bundle of other hoses and they should be semi-pointed away from the rear tire.

 

The Bike originally came out of Texas. And would the floats sticking cause it not to idle? I've checked the idle screw, its fine. I'm currently down to the carbs now, getting ready to pull them and clean them. Wish me luck!