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Category $\mathcal{O}$ for hybrid quantum groups and non-commutative Springer resolutions

Quan Situ

TL;DR

The paper develops a coherent model for the principal block ${oldsymbol{ m O}}^0_oldeta$ of the hybrid quantum group at a root of unity by relating it to the G-equivariant coherent sheaves on the non-commutative Springer resolution ${oldsymbol{ m A}}$ via an abelian equivalence ${oldsymbol{ m O}}^0_oldeta \\cong ext{Coh}^G(oldsymbol{ ilde{oldsymbol{ u}}}^*{oldsymbol{A}})$. It constructs a Steinberg-block equivalence and a de-equivariantized framework, then promotes these to a full equivalence ${oldsymbol{ m F}}$ by comparing images of Verma objects through deformations and base changes; this is extended to a $B$-equivariant setting, making ${oldsymbol{ m F}}$ an equivariantization of a refined equivalence ${oldsymbol{ m F}}^b$. The work further introduces a canonical grading on ${oldsymbol{ m O}}^0_oldeta$ and proves that graded Verma multiplicities are governed by generic Kazhdan–Lusztig polynomials, linking quantum-category structure to both Soergel-type and Beruh and Kazhdan–Lusztig theory. The results unify representation-theoretic and geometric perspectives, connect to the affine Hecke category via the non-commutative Springer resolution, and provide explicit graded data useful for SEO and search indexing of this topic. The framework advances the quantum analogue of the Kazhdan–Lusztig equivalence and yields new tools for understanding deformations, gradings, and equivariant structures in quantum group categories at roots of unity.

Abstract

The hybrid quantum group was firstly introduced by Gaitsgory, whose category $\mathcal{O}$ can be viewed as a quantum analogue of BGG category $\mathcal{O}$. We give a coherent model for its principal block at roots of unity, using the non-commutative Springer resolution defined by Bezrukavnikov--Mirković. In particular, the principal block is derived equivalent to the affine Hecke category. As an application, we endow the principal block with a canonical grading, and show that the graded multiplicity of simple module in Verma module is given by the generic Kazhdan--Lusztig polynomial.

Category $\mathcal{O}$ for hybrid quantum groups and non-commutative Springer resolutions

TL;DR

The paper develops a coherent model for the principal block of the hybrid quantum group at a root of unity by relating it to the G-equivariant coherent sheaves on the non-commutative Springer resolution via an abelian equivalence . It constructs a Steinberg-block equivalence and a de-equivariantized framework, then promotes these to a full equivalence by comparing images of Verma objects through deformations and base changes; this is extended to a -equivariant setting, making an equivariantization of a refined equivalence . The work further introduces a canonical grading on and proves that graded Verma multiplicities are governed by generic Kazhdan–Lusztig polynomials, linking quantum-category structure to both Soergel-type and Beruh and Kazhdan–Lusztig theory. The results unify representation-theoretic and geometric perspectives, connect to the affine Hecke category via the non-commutative Springer resolution, and provide explicit graded data useful for SEO and search indexing of this topic. The framework advances the quantum analogue of the Kazhdan–Lusztig equivalence and yields new tools for understanding deformations, gradings, and equivariant structures in quantum group categories at roots of unity.

Abstract

The hybrid quantum group was firstly introduced by Gaitsgory, whose category can be viewed as a quantum analogue of BGG category . We give a coherent model for its principal block at roots of unity, using the non-commutative Springer resolution defined by Bezrukavnikov--Mirković. In particular, the principal block is derived equivalent to the affine Hecke category. As an application, we endow the principal block with a canonical grading, and show that the graded multiplicity of simple module in Verma module is given by the generic Kazhdan--Lusztig polynomial.
Paper Structure (59 sections, 61 theorems, 296 equations)

This paper contains 59 sections, 61 theorems, 296 equations.

Table of Contents

  1. Introduction
  2. Hybrid quantum group and its category ${\mathcal{O}}$
  3. Main result
  4. Equivalence of Steinberg block and functor ${\mathbb V}$
  5. A de-equvariantization
  6. Deformations
  7. Kazhdan--Lusztig polynomial
  8. Organization of the paper
  9. Conventions
  10. Acknowledgements
  11. Quantum groups and their category ${\mathcal{O}}$
  12. Root data
  13. Affine Weyl group
  14. Quantum groups
  15. Harish-Chandra center
  16. ...and 44 more sections

Key Result

Theorem A

There is an equivalence of abelian categories intertwining the reflection functors on both sides.

Theorems & Definitions (108)

  • Theorem A: =Corollaries \ref{['cor 5.14']}&\ref{['cor 6.8']}
  • Theorem 1.1: Situ2
  • Proposition 1.2: =Proposition \ref{['prop 4.1']}
  • Theorem B: =Theorem \ref{['thm 5.13']}
  • Theorem C: =Theorem \ref{['thm 7.3']}
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Situ1
  • Lemma 2.4: BBASV and Situ1
  • ...and 98 more