Positivity conjectures for Kazhdan–Lusztig theory on twisted involutions: The finite case

Abstract

Let (W,S) be any Coxeter system and let w↦w⁎ be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have shown that the corresponding set of twisted involutions (i.e., elements w∈W with w−1=w⁎) naturally generates a module of the Hecke algebra of (W,S) with two distinguished bases. The transition matrix between these bases defines a family of polynomials Py,wσ which one can view as a “twisted” analogue of the much-studied family of Kazhdan–Lusztig polynomials of (W,S). The polynomials Py,wσ can have negative coefficients, but display several conjectural positivity properties of interest, which parallel positivity properties of the Kazhdan–Lusztig polynomials. This paper reports on some calculations which verify four such positivity conjectures in several finite cases of interest, in particular for the non-crystallographic Coxeter systems of types H3 and H4.