This may be the Gear Position Sensor, which is at the left rear of the engine block.
Do a search on GPS to find past threads. The GPS has been redesigned in newer bike. I have a 2008
I had this problem and after reading several threads here I did the following:
Remove GPS, one bolt I think, carefully preserve the spring and pin that will come out.
Clean up the oily mess.
Scotch-brite the little round brass contacts.
Reinstall.
I had to twist it while tightening (right or left I forget) to get the neutral light to come on.
No problems in 2 years.