[Paper Review] Harish-Chandra bimodules for quantized Slodowy slices

Victor Ginzburg|ArXiv.org|Jul 2, 2008

Algebraic structures and combinatorial models32 references15 citations

TL;DR

This paper introduces Harish-Chandra bimodules over Premet's quantized Slodowy slices, a noncommutative deformation of the Poisson algebra of polynomial functions on a Slodowy slice. By defining regular singularities via microlocal techniques and constructing translation functors, the author provides a category analogous to category O for these algebras, proves Premet's conjecture on primitive ideals, and constructs noncommutative resolutions via a noncommutative Proj-construction.

ABSTRACT

The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizaions of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, and to construct `noncommutative resolutions' of Slodowy slices via translation functors.

Motivation & Objective

• To address the lack of a category O analogue for Premet's quantized Slodowy slice algebras $A_c$, which are noncommutative deformations of Poisson algebras on Slodowy slices.
• To define a category of Harish-Chandra bimodules over $A_c$ that serves as a substitute for category O, using the notion of weak Harish-Chandra bimodules and regular singularities.
• To prove Premet's conjecture relating finite-dimensional $A_c$-modules to primitive ideals in $\mathcal{U}(\mathfrak{g})$ whose associated variety is the closure of the adjoint orbit of a nilpotent element $e$.
• To construct noncommutative resolutions of Slodowy slices via translation functors and a noncommutative Proj-construction, generalizing earlier results in noncommutative algebraic geometry.

Proposed method

• Introduces weak Harish-Chandra bimodules over filtered algebras $A$ with $\operatorname{gr} A$ commutative and Poisson, generalizing the classical notion for $\mathcal{U}(\mathfrak{g})$.
• Defines regular singularities for $A_c$-bimodules using microlocal techniques, extending results from $\mathscr{D}$-modules on flag varieties.
• Applies the Beilinson-Bernstein localization theorem to relate $\mathscr{D}$-modules on $\mathcal{B} \times \mathcal{B}$ to $({\mathcal{U}}_{\mu},{\mathcal{U}}_{\nu})$-bimodules, establishing equivalence between locally finite adjoint action and regular singularities.
• Constructs a Whittaker-type functor from Harish-Chandra $\mathcal{U}(\mathfrak{g})$-bimodules to Harish-Chandra $A_c$-bimodules, enabling transfer of representation-theoretic data.
• Uses translation functors on $A_c$-modules to define a noncommutative Proj-construction, yielding noncommutative resolutions of the Slodowy slice $\mathcal{S}$.
• Applies results on $G^{sc}$-equivariant $\mathscr{D}$-modules on $\mathcal{B} \times \mathcal{B}$ to deduce regular singularities, leveraging the fact that such modules have regular singularities due to finitely many orbits.

Experimental results

Research questions

• RQ1 Can a category of Harish-Chandra bimodules be defined for Premet's quantized Slodowy slice algebras $A_c$, serving as a replacement for category O?
• RQ2 What is the correct notion of regular singularities for $A_c$-bimodules, and how does it relate to the classical notion in $\mathscr{D}$-module theory?
• RQ3 Does the Whittaker functor construction from $\mathcal{U}(\mathfrak{g})$-bimodules to $A_c$-bimodules provide a new proof of Premet's conjecture on primitive ideals?
• RQ4 Can translation functors on $A_c$-modules be used to construct noncommutative resolutions of the Slodowy slice $\mathcal{S}$ via a noncommutative Proj-construction?
• RQ5 How does the category of Harish-Chandra $A_c$-bimodules relate to the geometry of the nilpotent orbit closure $\overline{\operatorname{Ad} G(e)}$?

Key findings

• The category of Harish-Chandra $A_c$-bimodules is defined as weak Harish-Chandra bimodules with regular singularities, providing a natural substitute for category O in the absence of a direct analogue.
• The Whittaker functor from $\mathcal{U}(\mathfrak{g})$-bimodules to $A_c$-bimodules is constructed and used to give a new, direct proof of Premet's conjecture relating finite-dimensional $A_c$-modules to primitive ideals with associated variety $\overline{\operatorname{Ad} G(e)}$.
• The equivalence between locally finite adjoint action and regular singularities for $\mathscr{D}$-modules on $\mathcal{B} \times \mathcal{B}$ is extended to $A_c$-bimodules via microlocal techniques.
• Translation functors on $A_c$-modules are defined and used to construct a noncommutative resolution of the Slodowy slice $\mathcal{S}$ via a noncommutative Proj-construction, generalizing earlier work by Gordon-Stafford and Boyarchenko.
• The construction of noncommutative resolutions via translation functors is expected to recover Boyarchenko's resolution in the subregular case, and is compatible with the geometry of Kleinian singularities via Brieskorn-Slodowy theory.
• The theory of Harish-Chandra bimodules over $A_c$ provides a framework for studying two-sided ideals and primitive ideals in $A_c$, generalizing the classical theory for $\mathcal{U}(\mathfrak{g})$.