Fourier-Orbit Construction of GKZ-Type Systems for Commutative Linear Algebraic Groups

Abstract: We introduce the Fourier-orbit construction of GKZ-type D-modules associated with commutative linear algebraic group actions G = TU (where T is an algebraic torus and U a unipotent group) on a vector space V. This framework generalizes the classical toric GKZ system to mixed torus-unipotent settings. We establish generic holonomicity via a parameter-free symbolic moment ideal, develop symbolic tools for rank analysis, and exhibit new families with Airy-type irregular behavior. Our approach recovers the classical GKZ rank formula in the pure torus case and provides explicit lower bounds in the mixed case.