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

Abstract

We study GKZ-type D-modules arising from the actions of commutative linear algebraic groups G = TU (where T is a torus and U is unipotent) on a vector space. Building on Hotta's equivariant D-module framework, we formalize a Fourier-orbit construction that recovers the classical toric GKZ system and extends it to mixed torus-unipotent settings. We prove generic holonomicity via a parameter-free symbolic moment ideal and introduce two symbolic tools - the tp-envelope and the symbolic cap - for effective rank analysis and, under mild regularity, exact rank computation. A torus slice yields an explicit lower bound by the normalized lattice volume, explaining sharpness in the pure torus case. Examples exhibit irregular (Airy-type) behavior and resonant non-holonomicity, highlighting new phenomena beyond the toric setting.