Hyundai Elantra 2005 air conditioner occasionally fails to come on: Why?

It's a great car. Recently, after I've been driving for twenty minutes or more, and then I stop and turn off the car, the air conditioner will fail to turn on when I start the car again. It will remain off for a few minutes. The AC control panel lights are on, but no air blows out. Then, it eventually comes on after a few minutes of driving, and it runs great. Plenty of cold air. No strange noises. Is something over-heating? Is it a wiring problem? Could it have something to do with fluids? Could a belt be going? Is it going to be expensive? 2005 Hyundai Elantra Experts, HELP!