Multidegree for bifiltered D-modules

In commutative algebra, E. Miller and B. Sturmfels defined the notion of multidegree for multigraded modules over a multigraded polynomial ring. We apply this theory to bifiltered modules over the Weyl algebra D. The bifiltration is a combination of the standard filtration by the order of differential operators and of the so-called V-filtration along a coordinate subvariety of the ambient space defined by M. Kashiwara. The multidegree we define provides a new invariant for D-modules. We investigate its relation with the L-characteristic cycles considered by Y. Laurent. We give examples from the theory of A-hypergeometric systems MA(β) defined by I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky. We consider the V-filtration along the origin. When the toric projective variety defined from the matrix A is Cohen–Macaulay, we have an explicit formula for the multidegree of MA(β).