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Categorical action for finite classical groups and its applications: characteristic 0

Pengcheng Li, Peng Shan, Jiping Zhang

TL;DR

The paper develops a categorical framework for finite classical groups by constructing a double quantum Heisenberg action on the representation category RG_\bullet-mod, yielding a Kac-Moody action of $\mathfrak{sl}'_{I_+}\oplus\mathfrak{sl}'_{I_-}$ with disjoint type A quivers on the whole category. It introduces colored weight functions $\mathbb{O}^+(u)$ and $\mathbb{O}^-(v)$ together with uniform projection to obtain complete invariants for irreducible modules in characteristic zero, including complete invariants for quadratic unipotent and unipotent modules. The theta correspondence is leveraged to fix signature ambiguities and to decategorify the action, enabling an explicit description of the Kac-Moody action on Grothendieck groups via first occurrences and modified Lusztig decompositions. Extra symmetries (spinor, determinant, diagonal automorphisms) enrich the categorification and refine the parametrization of irreducibles. Overall, the work provides a uniform, characteristic-free approach to Lusztig’s classification for finite classical groups and connects categorification with classical theta correspondences and Harish-Chandra theory, with future plans to extend to modular representations and block theory.

Abstract

In this paper, we construct a categorical double quantum Heisenberg action on the representation category of finite classical groups $\mathrm{O}_{2n+1}(q)$, $\mathrm{Sp}_{2n}(q)$ and $\mathrm{O}^{\pm}_{2n}(q)$ with $q$ odd. Over a field of characteristic zero or characteristic $\ell$ with $\ell\nmid q(q-1)$, we deduce a categorical action of a Kac-Moody algebra $\mathfrak{s}\mathfrak{l}'_{I_+}\oplus\mathfrak{s}\mathfrak{l}'_{I_-}$ on the representation category of finite classical groups. We show that the colored weight functions $\mathbb{O}^+(u)(\bullet)$, $\mathbb{O}^-(v)(\bullet)$ and uniform projection can distinguish all irreducible characters of finite classical groups. In particular, the colored weight functions are complete invariants of quadratic unipotent characters. We also show that using the theta correspondence and extra symmetries of categorical double quantum Heisenberg action, the Kac-Moody action on the Grothendieck group of the whole category can be determined explicitly.

Categorical action for finite classical groups and its applications: characteristic 0

TL;DR

The paper develops a categorical framework for finite classical groups by constructing a double quantum Heisenberg action on the representation category RG_\bullet-mod, yielding a Kac-Moody action of with disjoint type A quivers on the whole category. It introduces colored weight functions and together with uniform projection to obtain complete invariants for irreducible modules in characteristic zero, including complete invariants for quadratic unipotent and unipotent modules. The theta correspondence is leveraged to fix signature ambiguities and to decategorify the action, enabling an explicit description of the Kac-Moody action on Grothendieck groups via first occurrences and modified Lusztig decompositions. Extra symmetries (spinor, determinant, diagonal automorphisms) enrich the categorification and refine the parametrization of irreducibles. Overall, the work provides a uniform, characteristic-free approach to Lusztig’s classification for finite classical groups and connects categorification with classical theta correspondences and Harish-Chandra theory, with future plans to extend to modular representations and block theory.

Abstract

In this paper, we construct a categorical double quantum Heisenberg action on the representation category of finite classical groups , and with odd. Over a field of characteristic zero or characteristic with , we deduce a categorical action of a Kac-Moody algebra on the representation category of finite classical groups. We show that the colored weight functions , and uniform projection can distinguish all irreducible characters of finite classical groups. In particular, the colored weight functions are complete invariants of quadratic unipotent characters. We also show that using the theta correspondence and extra symmetries of categorical double quantum Heisenberg action, the Kac-Moody action on the Grothendieck group of the whole category can be determined explicitly.
Paper Structure (67 sections, 51 theorems, 307 equations, 4 tables)

This paper contains 67 sections, 51 theorems, 307 equations, 4 tables.

Table of Contents

  1. Overview
  2. Categorical actions
  3. Applications to classifications of irreducible modules in characteristic zero
  4. Theta correspondence
  5. Future work
  6. Organization of the paper
  7. Preliminaries
  8. Algebras and categories
  9. Quantum Heisenberg category
  10. From Heisenberg categorification to Kac-Moody categorification
  11. Double quantum Heisenberg category
  12. From Double quantum Heisenberg to $\mathfrak{s}\mathfrak{l}'_{I_+}\oplus\mathfrak{s}\mathfrak{l}'_{I_-}$-categorification
  13. Double quantum Heisenberg categorification for finite classical groups
  14. Categorical double quantum Heisenberg action
  15. Construction of classical groups
  16. ...and 52 more sections

Key Result

Theorem 1.3

Let $R=K$ or $k.$ Associated with $\mathcal{C}$, there is a $2$-representation $\Phi:\mathfrak{U}(\mathfrak{s}\mathfrak{l}'_{I})\rightarrow \mathfrak{Cat}_R$ defined on objects by $\lambda\mapsto \mathcal{C}_\lambda$, on generating 1-morphisms by $E_i 1_\lambda\mapsto E_i|_{\mathcal{C}_\lambda}$ and

Theorems & Definitions (103)

  • Definition 1.1: BSW1
  • Definition 1.2
  • Theorem 1.3: BSW2
  • Theorem 1.4: HL
  • Theorem 1.5: HL
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • ...and 93 more