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Singular Soergel bimodules

Geordie Williamson

TL;DR

The paper defines and analyzes a 2-category of singular Soergel bimodules as a natural extension of Soergel bimodules, and proves a full classification of indecomposable objects indexed by double cosets $W_I\backslash W / W_J$. It shows that the split Grothendieck group of this 2-category recovers the Schur algebroid, providing a categorification of its morphism spaces, and establishes that Soergel’s conjecture in characteristic zero would imply analogous character formulas for singular bimodules. The work develops standard and Bott–Samelson bimodules, introduces filtrations by nabla and delta flags, and constructs Demazure-operator–based infrastructure to study translation functors and extension-of-scalars; the resulting character theory connects to Kazhdan–Lusztig bases and yields positivity and duality properties. An erratum subsequently corrects a statement about Bott–Samelson cases, proposing fixes to the normalization of characters and dualities to preserve the categorification framework. Overall, the results deepen the link between Hecke-type categorifications, Schur algebroids, and singular representation-theoretic/geometric structures, with potential implications for singular blocks in category $\mathcal{O}$ and related categorifications.

Abstract

We define and study categories of singular Soergel bimodules, which are certain natural generalisations of Soergel bimodules. Indecomposable singular Soergel bimodules are classified, and we conclude that the split Grothendieck group of the 2-category of singular Soergel bimodules is isomorphic to the Schur algebroid. Soergel's conjecture on the characters of indecomposable Soergel bimodules in characteristic zero is shown to imply a similar conjecture for the characters of singular Soergel bimodules.

Singular Soergel bimodules

TL;DR

The paper defines and analyzes a 2-category of singular Soergel bimodules as a natural extension of Soergel bimodules, and proves a full classification of indecomposable objects indexed by double cosets . It shows that the split Grothendieck group of this 2-category recovers the Schur algebroid, providing a categorification of its morphism spaces, and establishes that Soergel’s conjecture in characteristic zero would imply analogous character formulas for singular bimodules. The work develops standard and Bott–Samelson bimodules, introduces filtrations by nabla and delta flags, and constructs Demazure-operator–based infrastructure to study translation functors and extension-of-scalars; the resulting character theory connects to Kazhdan–Lusztig bases and yields positivity and duality properties. An erratum subsequently corrects a statement about Bott–Samelson cases, proposing fixes to the normalization of characters and dualities to preserve the categorification framework. Overall, the results deepen the link between Hecke-type categorifications, Schur algebroids, and singular representation-theoretic/geometric structures, with potential implications for singular blocks in category and related categorifications.

Abstract

We define and study categories of singular Soergel bimodules, which are certain natural generalisations of Soergel bimodules. Indecomposable singular Soergel bimodules are classified, and we conclude that the split Grothendieck group of the 2-category of singular Soergel bimodules is isomorphic to the Schur algebroid. Soergel's conjecture on the characters of indecomposable Soergel bimodules in characteristic zero is shown to imply a similar conjecture for the characters of singular Soergel bimodules.

Paper Structure

This paper contains 35 sections, 61 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Applications of singular Soergel bimodules
  3. An overview of the classification
  4. Structure of this paper
  5. Coxeter groups and the Schur algebroid
  6. Coxeter groups
  7. The Hecke algebra
  8. The Schur algebroid
  9. Bimodules and homomorphisms
  10. Invariants, graphs and standard modules
  11. Reflection faithful representations and invariants
  12. Singular standard modules
  13. Enlarging the regular functions
  14. Standard modules and extension of scalars
  15. Support
  16. ...and 20 more sections

Key Result

Theorem 1

There is a natural bijection: More precisely, for every $p \in W_{I}\!\setminus \! W/W_{J}$ there exists a unique isomorphism class (up to shifts) of indecomposable bimodules $M \in {}^{I}\mathcal{B}^{J}$ whose support is ${}^{I}\! \operatorname{Gr}_{\le p}^{J}$.

Theorems & Definitions (119)

  • Theorem 1
  • Theorem 2
  • Conjecture 1: SoBimodules, Vermutung 1.13
  • Theorem 3
  • Proposition 2.1.1
  • Theorem 2.1.2: Kilmoyer
  • Theorem 2.1.3: Howlett
  • Proposition 2.1.4
  • Lemma 2.1.5
  • proof
  • ...and 109 more