Light leaves and Lusztig's conjecture

TL;DR

This paper introduces the double leaves basis for Hom spaces in Bott-Samelson-Soergel bimodules and provides a combinatorial algorithm to identify primes where Soergel's conjecture holds, impacting Lusztig's conjecture.

Contribution

It presents a new combinatorial basis and an algorithm linking Soergel's and Lusztig's conjectures, with applications to prime characterization and counterexamples.

Findings

Counterexamples to Lusztig's conjecture found by Williamson.

Algorithm determines primes for which Soergel's conjecture holds.

Applications include proofs of positivity of Kazhdan-Lusztig polynomial coefficients.

Abstract

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.