[Paper Review] On modular categories O for quantized symplectic resolutions

Ivan Losev|arXiv (Cornell University)|Dec 20, 2017

Algebraic structures and combinatorial models35 references3 citations

TL;DR

This paper establishes highest weight and standardly stratified structures on modular categories 𝒪 for quantized symplectic resolutions in large positive characteristic. It shows that these modular categories, constructed via reduction modulo p ≫ 0 of quantized algebras, recover the usual characteristic zero categories 𝒪 upon base change, and uses this framework to construct wall-crossing bijections and graded lifts, including Koszulity results for characteristic zero categories.

ABSTRACT

In this paper we study highest weight and standardly stratified structures on modular analogs of categories $\mathcal{O}$ over quantizations of symplectic resolutions and show how to recover the usual categories $\mathcal{O}$ (reduced mod $p\gg 0$) from our modular categories. More precisely, we consider a conical symplectic resolution that is defined over a finite localization of $\mathbb{Z}$ and is equipped with a Hamiltonian action of a torus $T$ that has finitely many fixed points. We consider algebras $\mathcal{A}_λ$ of global sections of a quantization in characterstic $p\gg 0$, where $λ$ is a parameter. Then we consider a category $ ilde{\mathcal{O}}_λ$ consisting of all finite dimensional $T$-equivariant $\mathcal{A}_λ$-modules. We show that for $λ$ lying in a {\it p-alcove} $\,^p\!A$, the category $ ilde{\mathcal{O}}_λ$ is highest weight (in some generalized sense). Moreover, we show that every face of $\,^p\!A$ that survives in $\,^p\!A/p$ when $p ightarrow \infty$ defines a standardly stratified structure on $ ilde{\mathcal{O}}_λ$. We identify the associated graded categories for these standardly stratified structures with reductions mod $p$ of the usual categories $\mathcal{O}$ in characteristic $0$. Applications of our construction include computations of wall-crossing bijections in characteristic $p$ and the existence of gradings on categories $\mathcal{O}$ in characteristic $0$.

Motivation & Objective

• To develop a modular framework for categories 𝒪 in positive characteristic for quantized symplectic resolutions.
• To establish highest weight and standardly stratified structures on these modular categories.
• To show that reductions modulo p ≫ 0 recover the classical characteristic zero categories 𝒪.
• To apply the modular construction to construct wall-crossing bijections and graded lifts in characteristic zero.
• To prove Koszulity of characteristic zero categories 𝒪 via specialization from modular categories.

Proposed method

• Constructs modular categories 𝒪 as T-equivariant finite-dimensional modules over algebras of global sections of quantizations in characteristic p ≫ 0.
• Identifies p-alcoves and uses compatible elements to define highest weight structures in the modular setting.
• Relies on translation bimodules and wall-crossing functors in the modular setting to relate different parameter regions.
• Uses reduction modulo p and base change to relate modular categories to classical characteristic zero categories 𝒪.
• Applies p-center techniques and Tor vanishing to lift structures from modular to characteristic zero.
• Employs graded lifts and Koszul duality techniques to transfer Koszulity from modular to characteristic zero categories.

Experimental results

Research questions

• RQ1 How can highest weight structures be defined in the modular setting for categories 𝒪 over quantized symplectic resolutions?
• RQ2 What is the relationship between modular categories 𝒪 in characteristic p ≫ 0 and the classical characteristic zero categories 𝒪?
• RQ3 How do standardly stratified structures in the modular setting correspond to those in characteristic zero?
• RQ4 Can wall-crossing functors in characteristic p be used to construct bijections between simple modules in different parameter regions?
• RQ5 Does the existence of Koszul graded lifts in modular categories imply Koszulity for the corresponding characteristic zero categories?

Key findings

• For λ in a p-alcove, the modular category 𝒪_λ is highest weight in a generalized sense under reasonable assumptions.
• Each face of the p-alcove that survives modulo p defines a standardly stratified structure on 𝒪_λ.
• The associated graded categories of these standardly stratified structures are isomorphic to reductions modulo p of the classical characteristic zero categories 𝒪.
• Wall-crossing functors in characteristic p induce bijections between simple modules, generalizing the characteristic zero case.
• Graded lifts of modular categories 𝒪 in characteristic p induce graded lifts in characteristic zero, and Koszulity is preserved under base change.
• If the modular category 𝒪 has Koszul graded lifts for infinitely many p, then the corresponding characteristic zero category 𝒪 is also Koszul.