  Let $(W,S)$ be any Coxeter system and let $w \mapsto w^*$ 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 $w \in
W$ 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 $P^\sigma_{y,w}$ which one can view as a
"twisted" analogue of the much-studied family of Kazhdan-Lusztig polynomials of
$(W,S)$. The polynomials $P^\sigma_{y,w}$ 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 $H_3$ and $H_4$.
