UV photography is indeed possible with digital cameras as well as with film. The sensor, despite its filters, is sufficiently sensitive to UV light for reflective UV photography.
There are two problems, though.
The first is the lenses. The glass lenses and their coatings absorb more than 99% of the UV light before it ever hits the sensor. There ARE a couple lenses that will transmit UV light. The most famous is a 105 mm lens with fluorite elements, made by Nikon, and it is almost exclusively used for forensic pathology. The other is made by Jenoptic. They cost thousands of dollars. Good luck finding one even at B&H let alone on the used market.
http://www.coastalopt.com …I_egczF-pMCFQGfHgodQUEfWg
The other problem is a filter. You need a filter that will cut off visible light, roughly anything longer than 350 nm. These are probably not made commercially for photography. You'd need to find an optical UV pass filter made for research purposes and have it physically adapted to mount on your lens.
It's a shame, because UV photography is gorgeous. I'd love to do it in 4x5 or medium format, but no lens exists for these formats.
But here's the one site that has a ton of info and pictures -- it's eye candy.
http://www.naturfotograf.com/uvstart.html