Bernstein inequality and holonomic modules

  title={Bernstein inequality and holonomic modules},
  author={Ivan V. Losev and Pavel Etingof},
  journal={arXiv: Representation Theory},
Finite dimensional Hopf actions on algebraic quantizations
Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [,], we showed that a semisimple Hopf action on a Weyl
Poisson traces, D-modules, and symplectic resolutions
The theory of Poisson traces (or zeroth Poisson homology) developed by the authors is surveyed, to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations.
Derived Equivalences for Symplectic Reflection Algebras
  • I. Losev
  • Mathematics
    International Mathematics Research Notices
  • 2019
In this paper we study derived equivalences for symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over these algebras and
Bernstein's inequality and holonomicity for certain singular rings
. In this manuscript, we prove the Bernstein inequality and develop the theory of holonomic D -modules for rings of invariants of finite groups in characteristic zero, and for strongly F -regular
Semisimplicity of the category of admissible D-modules
Using a representation theoretic parameterization for the orbits in the enhanced cyclic nilpotent cone, derived by the authors in a previous article, we compute the fundamental group of these orbits.
Non-vanishing of geometric Whittaker coefficients for reductive groups
We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GLn to general reductive groups. The key
An investigation into Lie algebra representations obtained from regular holonomic D-modules
Beilinson–Bernstein localisation [BB81] relates representations of a Lie algebra g to certain D-modules on the flag variety of g. In [Rom21], examples of sl2-representations which correspond to
Affine Springer Fibers, Procesi bundles, and Cherednik algebras
Let g be a semisimple Lie algebra, t its Cartan subalgebra and W the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology
Holonomic modules over Cherednik algebras, I
Quantum Hamiltonian Reduction for Polar Representations
Let G be a reductive complex Lie group with Lie algebra g and suppose that V is a polar G-representation. We prove the existence of a radial parts map rad : D(V ) → Aκ from the G-invariant


Quantizations of conical symplectic resolutions I: local and global structure
We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a
Cherednik algebras and differential operators on quasi-invariants
We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the
On primitive ideals
AbstractWe extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the Irreducibility theorem for associated varieties and Duflo theorem on primitive
Quantized symplectic actions and W -algebras
With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W-algebra of finite type. This algebra is obtained from the universal
On the category 𝒪 for rational Cherednik algebras
Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov
Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products
The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an
Poisson Traces and D-Modules on Poisson Varieties
To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of
Fundamental groups of symplectic singularities
Let (X, \omega) be an affine symplectic variety. Assume that X has a C^*-action with positive weights and \omega is homogeneous with respect to the C^*-action. We prove that the algebraic fundamental
Etingof’s conjecture for quantized quiver varieties
We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact