Why does the radome on the AWACS rotate?
Basically, the radar antenna of the AWACS system is just like this (source):



Rotating waveguide antenna (picture source)

The round shape is only an aerodynamic fairing. Proof:



Boeing E-3A AWACS cutaway (picture source)

A surveillance antenna can also be fixed. Electronic beam steering helps to view a sector of approximately ±60°, but in order to get a 360° sweep the airplane needs to fly circles. Beam steering is also used on the rotating antenna to rapidly switch between several targets, so this is not unique to a fixed antenna:



Pakistani AWACS based on EMB-145 (picture source)