Thomas Gobet (Nancy), Positivity properties in Hecke algebras of arbitrary Coxeter groups.

Abstract:

The Iwahori-Hecke algebra of a Coxeter group has a standard and a costandard basis, as well as two canonical bases introduced by Kazhdan and Lustzig. In 1979, they conjectured that the coefficients of the canonical basis in the standard basis are all positive. This conjecture was proven recently in full generality by Elias and Williamson, following an approach of Soergel.

In his 1987 thesis, Dyer conjectured generalizations of the Kazhdan-Lusztig positivity. More precisely, he conjectured that the product of an element of the canonical basis with an element of the standard basis has positive coefficients in the standard basis. He also conjectured a natural "inverse" positivity statement. These conjectures were proven by Dyer and Lehrer in 1990 for finite Weyl groups using the geometry of the flag variety of the corresponding algebraic group, and then by Dyer for finite Coxeter groups (partially unpublished). Using Dyer's notion of biclosed sets of roots, we consider a family of bases "twisted" by these biclosed sets, containing both the standard and costandard bases and show that an element of the canonical basis has a positive expansion in any basis from this family. The key tool is to consider twisted filtrations of Soergel bimodules (these bimodules categorify the canonical basis of the Hecke algebra) and interpret the coefficients as multiplicities in these filtrations. This statement implies Dyer's conjecture in full generality. We then consider the "inverse" positivity statement. In that case, one has to consider the bounded homotopy category of Soergel bimodules to obtain natural categorifications of the elements of the twisted standard bases. In case the biclosed set is finite or cofinite, we show that the corresponding twisted standard basis has positive coefficients in the canonical basis, implying Dyer's inverse positivity conjecture.

Elements of the twisted standard bases turn out to be images of an interesting family of elements of the corresponding Artin-Tits or braid group of the Coxeter system called "Mikado braids". If time allows, we will mention possible relations with representation theory of braid groups.