Relative hard Lefschetz for Soergel bimodules

Abstract

We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan–Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category associated by Lusztig to any two-sided cell in a Coxeter group is rigid and pivotal.