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Relative Serre duality for Hecke categories

Quoc P. Ho, Penghui Li

TL;DR

This work proves a general Serre duality-type relation for Hecke categories: for any connected reductive group $G$ and a parabolic $P$ with Levi $L$, the left and right parabolic induction adjoints are related by the relative full twist, namely $\iota^R \simeq \iota^L(\mathrm{FT}_{G,L} \star -)$ where $\mathrm{FT}_{G,L}=\mathrm{FT}_L^{-1}\star\mathrm{FT}_G$. The authors execute a geometric strategy using the geometric Hecke category $\mathcal{H}_G$ (graded sheaves on $B\backslash G/B$) and translate between the geometric and algebraic pictures via an equivalence with the Soergel bimodule category. The proof hinges on two key components: (i) showing $\mathrm{FT}_G\star-$ induces an equivalence between kernel categories $\ker\iota^R$ and $\ker\iota^L$, and (ii) constructing a morphism $\alpha: \mathrm{FT}_G \to \mathrm{FT}_L$ with $\iota^L(\alpha)$ an equivalence; this is achieved by factoring $w_0$ through $W_L$ and exploiting standard-to-costandard morphisms and Bruhat decompositions. The result generalizes earlier type-A Serre duality to arbitrary connected reductive groups, providing a robust geometric route to understanding adjoint relationships in Hecke categories and their topological applications.

Abstract

We prove a conjecture of Gorsky, Hogancamp, Mellit, and Nakagane in the Weyl group case. Namely, we show that the left and right adjoints of the parabolic induction functor between the associated Hecke categories of Soergel bimodules differ by the relative full twist.

Relative Serre duality for Hecke categories

TL;DR

This work proves a general Serre duality-type relation for Hecke categories: for any connected reductive group and a parabolic with Levi , the left and right parabolic induction adjoints are related by the relative full twist, namely where . The authors execute a geometric strategy using the geometric Hecke category (graded sheaves on ) and translate between the geometric and algebraic pictures via an equivalence with the Soergel bimodule category. The proof hinges on two key components: (i) showing induces an equivalence between kernel categories and , and (ii) constructing a morphism with an equivalence; this is achieved by factoring through and exploiting standard-to-costandard morphisms and Bruhat decompositions. The result generalizes earlier type-A Serre duality to arbitrary connected reductive groups, providing a robust geometric route to understanding adjoint relationships in Hecke categories and their topological applications.

Abstract

We prove a conjecture of Gorsky, Hogancamp, Mellit, and Nakagane in the Weyl group case. Namely, we show that the left and right adjoints of the parabolic induction functor between the associated Hecke categories of Soergel bimodules differ by the relative full twist.

Paper Structure

This paper contains 17 sections, 10 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Soergel bimodules
  3. Parabolic induction and restriction functors
  4. Serre duality for Hecke categories
  5. Main result
  6. Geometric Hecke categories
  7. Geometric setup
  8. Standard and co-standard objects
  9. Geometric parabolic induction and restriction functors
  10. Parabolic induction functor
  11. Parabolic restriction functors
  12. The proof
  13. The strategy
  14. Proof of \ref{['serre_duality_hecke:prop:FT_G_L_induces_equiv_of_cats']}
  15. Proof of \ref{['serre_duality_hecke:prop:FT_to_FT_L']}
  16. ...and 2 more sections

Key Result

Theorem 1.3.1

For $G = \mathop{\mathrm{GL}}\nolimits_n$, $P$ the parabolic subgroup given by the partition $(r, 1, 1, \dots, 1)$ (and hence, $L \simeq \mathop{\mathrm{GL}}\nolimits_r \times \mathbb{G}_\text{\normalfont\sffamilym}^{n-r}$), we have a natural equivalence of functors $\iota^R \simeq \iota^L(\text{\no

Theorems & Definitions (17)

  • Theorem 1.3.1: gorsky_serre_2019
  • Theorem 1.4.1
  • Theorem 2.1.2: ho_revisiting_2022
  • Remark 2.1.3
  • Proposition 3.1.1
  • Proposition 3.1.2
  • proof : Proof of \ref{['serre_duality_hecke:thm:main']}
  • Lemma 3.2.1
  • proof
  • Lemma 3.2.2
  • ...and 7 more