Hypergeometric systems from groups with torsion

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 [13] 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, [6] , [2] .