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

Abstract

Let $(W,S)$ be a Coxeter system and let $w\mapsto w^{\ast }$ be an involution of $W$ which preserves the set of simple generators $S$. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements $w\in W$ with $w^{-1}=w^{\ast }$) 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 $P_{y,w}^{\sigma }$ which one can view as "twisted" analogues of the much-studied Kazhdan-Lusztig polynomials of $(W,S)$. The polynomials $P_{y,w}^{\sigma }$ can have negative coefficients, but display several conjectural positivity properties of interest. This paper reviews Lusztig's construction and then proves three such positivity properties for Coxeter systems which are universal (i.e., having no braids relations), generalizing previous work of Dyer. Our methods are entirely combinatorial and elementary, in contrast to the geometric arguments employed by Lusztig and Vogan to prove similar positivity conjectures for crystallographic Coxeter systems.

30 pages; v2: some corrections, revisions, updated references; v3: some minor further corrections and reference updates; v4: minor correction to part (c) of proof of Theorem 3.22, some typos fixed, final version

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• Quasisplit Hecke algebras and symmetric spaces
• Asymptotic Hecke algebras and involutions
• The Hodge theory of Soergel bimodules
• A $(-q)$-analogue of weight multiplicities
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• Involution words II: braid relations and atomic structures
• Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures
• Schur P-positivity and involution Stanley symmetric functions
• A symplectic refinement of shifted Hecke insertion
• On involutions in Weyl groups
• Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the finite case
• On involutions in symmetric groups and a conjecture of Lusztig
• Comparing and characterizing some constructions of canonical bases from Coxeter systems
• Affine transitions for involution Stanley symmetric functions
• Integral $u$-deformed involution modules