New bases of some Hecke algebras via Soergel bimodules

For extra-large Coxeter systems (m(s,r)>3), we construct a natural and explicit set of Soergel bimodules D={Dw}w∈W such that each Dw contains as a direct summand (or is equal to) the indecomposable Soergel bimodule Bw. When decategorified, we prove that D gives rise to a set {dw}w∈W that is actually a basis of the Hecke algebra. This basis is close to the Kazhdan–Lusztig basis and satisfies a positivity condition.