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Let $(W,S)$ be a Coxeter system and let $w \mapsto w^*$ be an involution of$W$ which preserves the set of simple generators $S$. Lusztig and Vogan haverecently shown that the 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 basesdefines a family of polynomials $P^\sigma_{y,w}$ which one can view as"twisted" analogues of the much-studied Kazhdan-Lusztig polynomials of $(W,S)$.The polynomials $P^\sigma_{y,w}$ can have negative coefficients, but displayseveral conjectural positivity properties of interest. This paper reviewsLusztig's construction and then proves three such positivity properties forCoxeter systems which are universal (i.e., having no braids relations),generalizing previous work of Dyer. Our methods are entirely combinatorial andelementary, in contrast to the geometric arguments employed by Lusztig andVogan to prove similar positivity conjectures for crystallographic Coxetersystems.

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case