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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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