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Irregular hypergeometric $$\mathcal D$$-modules. (English) Zbl 1236.14018
The paper under review studies the irregularity of the hypergeometric $$\mathcal{D}$$-modules $$\mathcal{M}_A(\beta)$$ introduced by I. M. Gel’fand, M. I. Graev, M. M. Kapranov and A. V. Zelevinsky using explicit construction of Gevrey series along coordinate subspaces of $$\mathbb{C}^n$$. Here $$A$$ denotes a full rank $$d\times n$$-matrix with integer entries and $$\beta=(\beta_1,\dots,\beta_d)$$ are complex parameters. The module $$\mathcal{M}_A(\beta)$$ is the quotient $$\mathcal{D}/H_A(\beta)$$ where $$H_A(\beta)$$ is made of toric operators and Euler operators.
After reviewing facts on Gevrey series and slopes of $$\mathcal{D}$$-modules, the author explicitly constructs Gevrey series solutions of $$\mathcal{M}_A(\beta)$$ along coordinate subspaces. Using those series, ideas from M. Schulze and U. Walther [Duke Math. J. 142, No. 3, 465–509 (2008; Zbl 1144.13012)] and the Comparison Theorem of the slopes of Y. Laurent and Z. Mebkhout [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 1, 39–69 (1999; Zbl 0944.14007)], she then gives in Theorem 5.9 a combinatorial description of the set of slopes of $$\mathcal{M}_A(\beta)$$ along coordinate hyperplanes. This generalizes the result of Schulze and Walther (in the case of coordinate hyperplanes) because there are no further assumptions on $$A$$ (for instance, pointedness).
The author also gives lower bounds for the dimensions of the Gevrey solution spaces, which are the actual dimensions for very generic parameters. Under some conditions on $$A$$ and $$\beta$$, she finally obtains a basis of the stalk at a generic point of the irregularity sheaves of $$\mathcal{M}_A(\beta)$$ along a coordinate hyperplane.

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules 13N10 Commutative rings of differential operators and their modules 33C99 Hypergeometric functions 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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