I heard that $D$-modules on a quotient stack $X/G$, where $X$ is a variety and $G$ is an algebraic group, are equivalent to strongly $G$-equivariant $D$-modules on $X$. (See, e.g. D. Gaitsgory's NOTES ON 2D CONFORMAL FIELD THEORY AND STRING THEORY pp.5)
I wonder if we can have a more explicit description of $D$-modules on the quotient stack $X/G$ when $X$ is a smooth curve and $G$ is a finite group scheme.