A non-perverse Soergel bimodule in type A

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.