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.
Cite
@article{arxiv.2509.03881,
title = {Fourier-Orbit Construction of GKZ-Type Systems for Commutative Linear Algebraic Groups},
author = {Go Okuyama},
journal= {arXiv preprint arXiv:2509.03881},
year = {2025}
}
Comments
We found a serious flaw in the proof of Proposition 5.18, affecting the validity of the main result on the surjection onto the diagonal GKZ system. The current argument does not ensure well-definedness with unipotent contributions. We will revise and provide a corrected approach in a future version