A non-perverse Soergel bimodule in type A

TL;DR

This paper constructs indecomposable Soergel bimodules in type A with non-zero endomorphisms of negative degree, revealing non-perverse parity sheaves and challenging previous assumptions about their structure.

Contribution

It demonstrates the existence of non-perverse parity sheaves in type A by constructing indecomposable Soergel bimodules with negative degree endomorphisms.

Findings

Existence of indecomposable Soergel bimodules with negative degree endomorphisms in type A

Counterexamples to expected bounds in Lusztig's conjecture

Identification of non-perverse parity sheaves in type A

Abstract

A basic question concerning indecomposable Soergel bimodules is to understand their endomorphism rings. In characteristic zero all degree-zero endomorphisms are isomorphisms (a fact proved by Elias and the second author) which implies the Kazhdan-Lusztig conjectures. More recently, many examples in positive characteristic have been discovered with larger degree zero endomorphisms. These give counter-examples to expected bounds in Lusztig's conjecture. Here we prove the existence of indecomposable Soergel bimodules in type A having non-zero endomorphisms of negative degree. This gives the existence of a non-perverse parity sheaf in type A.