Graded Holonomic D-modules on Monomial Curves

Abstract: In this paper, we study the holonomic $D$-modules when $D$ is the ring of $k$-linear differential operators on $A = k[\Gamma]$, the coordinate ring of an affine monomial curve over the complex numbers $k = \mathbb C$. In particular, we consider the graded case, and classify the simple graded $D$-modules and compute their extensions. The classification over the first Weyl algebra $D = A_1(k)$ is obtained as a special case.