Holonomic modules for rings of invariant differential operators

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals [Formula: see text].

On Some Topological Properties of Fourier Transforms of Regular Holonomic -Modules

We study Fourier transforms of regular holonomic ${\mathcal{D}}$-modules. In particular, we show that their solution complexes are monodromic. An application to direct images of some irregular holonomic ${\mathcal{D}}$-modules will be given. Moreover, we give a new proof of the classical theorem of Brylinski and improve it by showing its converse.

RIEMANN–HILBERT CORRESPONDENCE FOR MIXED TWISTOR -MODULES

We introduce the notion of regularity for a relative holonomic ${\mathcal{D}}$-module in the sense of Monteiro Fernandes and Sabbah [Internat. Math. Res. Not. (21) (2013), 4961–4984]. We prove that the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes is essentially surjective by constructing a right quasi-inverse functor. When restricted to relative ${\mathcal{D}}$-modules underlying a regular mixed twistor ${\mathcal{D}}$-module, this functor satisfies the left quasi-inverse property.

A boundedness theorem for nearby slopes of holonomic -modules

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic${\mathcal{D}}$-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections${\mathcal{E}}_{t}$parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the${\mathcal{E}}_{t}$. This generalizes the regularity of the Gauss–Manin connection proved by Griffiths, Katz and Deligne.

MODULES OVER QUANTUM LAURENT POLYNOMIALS

We show that the Gelfand–Kirillov dimension for modules over quantum Laurent polynomials is additive with respect to tensor products over the base field. We determine the Brookes–Groves invariant associated with a tensor product of modules. We study strongly holonomic modules and show that there are nonholonomic simple modules.