The finned box is the voltage regulator/rectifier.
While they can get hot (reason for cooling fins), I don't think they should get super hot.
They have been known to fail due to overheating, causing either no current to the battery or over-charging of the battery.
I'd grab a digital ohmmeter, and with the bike switched on and running, and with high-beam on, check the voltage across the battery posts with the bike at 4500 RPM.
You should get around 14.5V DC. It should NOT exceed 15.5V DC.
Last edited by SubSailor; 08-04-2012 at 02:35 PM.