Are lower-case-g equivariant D-modules more like imposing equivariance with respect to a simply connected group or an adjoint group? SL_n (even SL_2) has a center, so there is always some equivariant local system on the regular orbit. In SL_2, this regular orbit is topologically C x C^*, and I think that SL_2 equivariance means that the monodromy around that C^* squares to 1. I don't know what happens for GL_n or PGL_n.