Elsevier

Advances in Mathematics

Volume 308, 21 February 2017, Pages 941-963
Advances in Mathematics

Bernstein inequality and holonomic modules

https://doi.org/10.1016/j.aim.2016.12.033Get rights and content
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Abstract

In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, and spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows one to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.

MSC

16W70

Keywords

Filtered algebras
Symplectic leaves
Bernstein inequality
Holonomic modules
Symplectic resolutions
Symplectic reflection algebras
Quantum Hamiltonian reductions

Cited by (0)

The paper contains an appendix, which is joint with Pavel Etingof from MIT.