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Center of the category $\mathcal{O}$ for a hybrid quantum group

Quan Situ

TL;DR

The paper proves that the center Z(O) of the category O for a hybrid quantum group U^hb_ζ at a root of unity is isomorphic to the completed cohomology H^•(Gr^ζ)^∧ of the ζ-fixed locus in the affine Grassmannian, extending a deformed version from prior work. It further establishes an abelian equivalence for the Steinberg block O^{[-ρ]} with Coh^G(T^*(G/B)) by passing to a central extension at q=1 and employing the Arkhipov–Bezrukavnikov–Ginzburg correspondence. The principal and singular blocks are analyzed via translation functors and trace constructions, allowing the transfer of center information across blocks and yielding isomorphisms with the corresponding flag-variety cohomologies. Collectively, these results unify algebraic centers of quantum-group categorical representations with geometric objects on affine Grassmannians and Springer resolutions, enriching the geometric representation-theory framework for quantum groups at roots of unity.

Abstract

We establish an algebra isomorphism between the center of the category $\mathcal{O}$ for a hybrid quantum group at a root of unity $ζ$ and the cohomology of $ζ$-fixed locus on affine Grassmannian. A deformed version of this isomorphism was established in the previous paper of the author. For the Steinberg block of $\mathcal{O}$, we construct an abelian equivalence to the category of equivariant sheaves on the Springer resolution.

Center of the category $\mathcal{O}$ for a hybrid quantum group

TL;DR

The paper proves that the center Z(O) of the category O for a hybrid quantum group U^hb_ζ at a root of unity is isomorphic to the completed cohomology H^•(Gr^ζ)^∧ of the ζ-fixed locus in the affine Grassmannian, extending a deformed version from prior work. It further establishes an abelian equivalence for the Steinberg block O^{[-ρ]} with Coh^G(T^*(G/B)) by passing to a central extension at q=1 and employing the Arkhipov–Bezrukavnikov–Ginzburg correspondence. The principal and singular blocks are analyzed via translation functors and trace constructions, allowing the transfer of center information across blocks and yielding isomorphisms with the corresponding flag-variety cohomologies. Collectively, these results unify algebraic centers of quantum-group categorical representations with geometric objects on affine Grassmannians and Springer resolutions, enriching the geometric representation-theory framework for quantum groups at roots of unity.

Abstract

We establish an algebra isomorphism between the center of the category for a hybrid quantum group at a root of unity and the cohomology of -fixed locus on affine Grassmannian. A deformed version of this isomorphism was established in the previous paper of the author. For the Steinberg block of , we construct an abelian equivalence to the category of equivariant sheaves on the Springer resolution.
Paper Structure (63 sections, 45 theorems, 240 equations)

This paper contains 63 sections, 45 theorems, 240 equations.

Table of Contents

  1. Introduction and notations
  2. Main results
  3. Center of category ${\mathcal{O}}$ for $U^\mathrm{hb}_\zeta$
  4. Equivalence for Steinberg block
  5. Main steps of proof
  6. Center of Steinberg block
  7. Center of principal block
  8. Center of singular blocks
  9. Arrangement of the paper
  10. Notations and conventions
  11. Notations
  12. Root data
  13. Rings
  14. Affine Weyl group
  15. Affine flag varieties
  16. ...and 48 more sections

Key Result

Theorem A

There is an algebra isomorphism Under the map $\overline{\mathbf{b}}$, the decomposition (equ 1.00) is compatible with the block decomposition ${\mathcal{O}}=\bigoplus\limits_\omega {\mathcal{O}}^{[\omega]}$ labelled by the singular type $\omega$. In other words, $\overline{\mathbf{b}}$ restricts to isomorphisms for each parahoric/singular type $\omega$.

Theorems & Definitions (84)

  • Theorem A: Theorems \ref{['prop 4.12']}, \ref{['thm 5.10']} and \ref{['thm 5.17']}
  • Theorem B: Theorem \ref{['thm 4.7']} and Corollary \ref{['cor 4.8']}
  • Lemma 2.1: Situ1
  • Lemma 2.2: BBASV and Situ1
  • Lemma 2.3: Situ1
  • Theorem 2.4: Situ1
  • Theorem 3.1: DeCKP92
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 74 more