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

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.