There's a few things to try, which may/may not help (or may take a combo)
See if you can force your iPad to reboot. Hold down the sleep and home keys for 30 seconds. If you see a silver apple it's rebooted and may come back.
It's possible that your battery is totally dead. if that's the case then you may need to leave it plugged in for an hour or more to see some sign of life.
Is the cable good? Is the plug you're using good? If you have others, give it a try with them. (And if you're trying to revive the long discharged you may need to let it set for a few hours)
Your battery may be shot. If your device is still under warranty and they determine that it's a manufacturer's defect, Apple may replace it for free. If your device isn't still under warranty, then you can still get the battery replaced, you'll just need to pay for it.
As to it saying 'not supported', iOS6 I believe brought with it software that looks for signed chips in the lightning cables. Basically if a company pays for permission to make accessories, they get the codes for a signed chip that's in the cable, and the iOS software looks for those chips. If your cable happened to be a non-authorized cable, then that could account for the 'not compatible' warning.
You can try to troubleshoot yourself, or you can make an appointment at a local Apple store or authorized repair place and have them help you with the troubleshooting.