The most likely cause is you've got an idle problem. When the engine speed drops the alternator + battery is not enough to keep up with the high electrical load and the lights should dim.

The compressor puts a relatively high demand on the engine in idle and so marginal problems can become more obvious.

As there are quite a few problems that can cause rough idle troubleshooting can take a while so be prepared to spend time or money finding the cause. Personally I would start with cleaning the MAF (a good idea in any case if you haven't done it in a while), then check the fuel filter. If these are the original plugs and wires it is time to change them anyway. If they've been replaced look for loose wires. Once past these items then check the fuel pump.

If you still haven't found a cause I would purchase one of the better OBD scanners that will both record and display multiple sensor readings simultaneously, then leave it hooked up until you duplicate the problem.

Good luck!