Hypergeometric systems from groups with torsion

Abstract

We consider $A$-hypergeometric (or GKZ-)systems in the case where the grading (character) group is an arbitrary finitely generated Abelian group. Emulating the approach taken for classical GKZ-systems in arXiv:math/0406383 that allows for a coefficient module, we show that these $D$-modules are holonomic systems. For this purpose we formulate an Euler--Koszul complex in this context, built on an extension of the category of $A$-toric modules. We derive that these new systems are regular holonomic under circumstances that are similar to those that lead to regular holonomic classical GKZ-systems. For the appropriate coefficient module, our $D$-modules specialize to the "better behaved GKZ-systems" introduced by Borisov and Horja. We certify the corresponding $D$-modules as regular holonomic, and establish a holonomic duality on the level of $D$-modules that was suggested on the level of solutions by Borisov and Horja and later shown by Borisov and Han in a special situation (arXiv:1308.2238, arXiv:2301.01374).