I've been trying various groundcovers, mostly those native to the mid-Atlantic region, for different parts of my yard. One of the tough spots is on a slope under a large silver maple. It gets some eastern sun in the morning, but is mostly dry shade. The Pennsylvania sedge (Carex pensylvanica) I planted a couple of years ago is doing well and starting to spread. It is a pretty little grass-like plant that grows about six inches tall.
Last year I added Sedum ternatum, also native around here, and it has come through the winter well.
At the native plant sale at the National Arboretum, I picked up some Pachysandra procumbens after one of the vendors told me her sister had good success growing it under a large tree. I believe it is native a little further south than where I live in Maryland, but it will sure beat the non-native pachysandra, which I have plenty of in other parts of the yard. It's still dormant right now, so I'll have to see how it does.