2025/09/04 by Go Okuyama, Okuyama, Go
Mathematics · #14L30 #14M25 #32C38 (Primary) #33C70 (Secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2509.03881
openalex publication_date 2025/09/04 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28
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.