Singular Soergel bimodules

TL;DR

This paper introduces and analyzes categories of singular Soergel bimodules, extending the theory of Soergel bimodules and connecting their characters to Schur algebroids, with implications for conjectures in representation theory.

Contribution

It defines singular Soergel bimodules, classifies indecomposables, and relates their Grothendieck group to the Schur algebroid, extending Soergel's conjecture.

Findings

Classification of indecomposable singular Soergel bimodules

Isomorphism of Grothendieck group with Schur algebroid

Implication of Soergel's conjecture for singular bimodules

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.