  We introduce the Double leaves basis, a combinatorial basis for the Hom
spaces between two Bott-Samelson-Soergel bimodules. As an application we give a
combinatorial algorithm to find, for any given Weyl or affine Weyl group, the
set of primes for which Soergel's conjecture hold. This conjecture for Weyl
groups is equivalent to a part of Lusztig's conjecture and for affine Weyl
groups implies (and is probably equivalent to) the full Lusztig conjecture.
Following this double leaves approach G. Williamson found counterexamples to
Lusztig's conjecture. The double leaves basis has found other spectacular
applications in the recent proof by B. Elias and G. Williamson of the
positivity of the coefficients of Kazhdan-Lusztig polynomials for any Coxeter
system and in their algebraic proof of Kazhdan-Lusztig conjecture.
