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Tautological systems and free divisors. (English) Zbl 1451.14054
The paper under review deals with systems of differential equations defined by certain prehomogeneous vector spaces endowed with actions of algebraic groups admitting open dense orbits. Such $$\mathcal D$$-modules can be considered as examples of the so-called tautological systems studied in many works (see [M. Kapranov, in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30–July 4, 1997, and in Kyoto, Japan, July 7–11 1997. Singapore: World Scientific. 236–281 (1998; Zbl 0987.33008); B. H. Lian et al., J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457–1483 (2013; Zbl 1272.14033)]).
In fact, the authors investigate the case of reductive groups whose orbits have complements which are linear free divisors satisfying the strongly Koszul condition [M. Granger and M. Schulze, Publ. Res. Inst. Math. Sci. 46, No. 3, 479–506 (2010; Zbl 1202.14046)]. Under these assumptions it is proved that the associated tautological systems underlie mixed Hodge modules. Moreover, the authors give an explicit representation of the corresponding $$\mathcal D$$-modules similarly to the case of GKZ-systems [T. Reichelt, Compos. Math. 150, No. 6, 911–941 (2014; Zbl 1315.14016)].
##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects) 14M17 Homogeneous spaces and generalizations
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