  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.
