Lusztig's conjectures P1–P15 for Hecke algebras of Coxeter groups with arbitrary positive weight functions

For a Coxeter system $(W,S)$ of finite rank equipped with a positive weight function $L$, Lusztig's monograph 'Hecke algebras with unequal parameters' poses fifteen structural conjectures P1–P15 for the Kazhdan–Lusztig basis of the associated Iwahori–Hecke algebra, concerning the $a$-function, the leading $\gamma$-coefficients, the two-sided cells, and the well-definedness of the asymptotic Hecke algebra $J$. Boundedness of every positively weighted finite-rank Coxeter group was proved in 2025 (Chen–Hu), which, combined with Elias–Williamson's proof of Kazhdan–Lusztig positivity, settles the equal-parameter and quasisplit cases. The problem asks whether P1–P15 hold for every positively weighted finite-rank Coxeter system; the surviving open core is the case of general (non-quasisplit) unequal parameters, where no positivity mechanism is known.

Background

A Coxeter system $(W,S)$ is a group generated by a finite set $S$ of involutions with relations $(st)^{m(s,t)}=1$; examples range from finite Weyl groups to affine Weyl groups and to infinite non-crystallographic reflection groups. A positive weight function is a map $L\colon W\to\mathbb{N}$ with $L(w)>0$ for $w\neq 1$ and $L(ww')=L(w)+L(w')$ whenever the word lengths add; the choice $L=\ell$ (the length function) is called the equal-parameter case. The Iwahori–Hecke algebra $H$ is the free module over $A=\mathbb{Z}[v^{\pm 1}]$ with basis $\{T_w\}$ and multiplication determined by $T_s^2=(v^{L(s)}-v^{-L(s)})T_s+1$. It carries the Kazhdan–Lusztig basis $\{c_w\}$, characterized by a bar involution $v\mapsto v^{-1}$ and a triangularity condition. Writing $T_xT_y=\sum_z f_{x,y,z}T_z$ and $c_xc_y=\sum_z h_{x,y,z}c_z$ with $f,h\in A$, the pair $(W,S,L)$ is called bounded if some $N\geq 0$ has $v^{-N}f_{x,y,z}\in\mathbb{Z}[v^{-1}]$ for all $x,y,z$; boundedness defines Lusztig's $a$-function, $a(z)$ being the smallest integer with $v^{-a(z)}h_{x,y,z}\in\mathbb{Z}[v^{-1}]$ for all $x,y$, and the leading coefficients $\gamma_{x,y,z^{-1}}$ via $h_{x,y,z}\equiv\gamma_{x,y,z^{-1}}v^{a(z)}\bmod v^{a(z)-1}\mathbb{Z}[v^{-1}]$. Two-sided Kazhdan–Lusztig cells are defined via preorders on $W$: $x\leq_L y$ if $c_x$ appears in $hc_y$ for some $h\in H$, $x\leq_R y$ if $c_x$ appears in $c_yh$ for some $h\in H$, and $x\leq_{LR} y$ if $c_x$ appears in $hc_yh'$ for some $h,h'\in H$; the two-sided cells are the equivalence classes of $x\sim_{LR}y$ (that is, $x\leq_{LR}y$ and $y\leq_{LR}x$), and they partition $W$ into the blocks that the conjectures control. In his monograph Lusztig posed, for any positively weighted finite-rank Coxeter system, the conjectures P1–P15 (arXiv:math/0208154, §14.2), under the boundedness assumption: they concern the $a$-function, the leading $\gamma$-coefficients, and the two-sided cells, and assert that the free abelian group $J$ with basis $\{t_w\}$ and multiplication $t_xt_y=\sum_z\gamma_{x,y,z^{-1}}t_z$ is a well-defined associative algebra with unit — the asymptotic Hecke algebra, whose based rings describe cell representations. For finite Coxeter groups, Geck constructed unconditionally a modified asymptotic algebra $\tilde J$ (arXiv:1110.5672, §3); for such $W$, if P1–P15 hold, then $J$ exists and $J=\tilde J$ (Remark 3.14). Before 2025 the conjectures were known only in restricted classes: for $L=\ell$ conditional on boundedness, via Elias–Williamson's proof of the Kazhdan–Lusztig positivity conjecture by the Hodge theory of Soergel bimodules (algebraic categories of bimodules on the reflection representation whose intersection forms control Kazhdan–Lusztig polynomials); in the quasisplit weighted case (weights permuted by diagram automorphisms), again conditional on boundedness; for rank-2 affine Weyl groups with arbitrary parameters (Guilhot–Parkinson); for all rank-3 Coxeter groups including hyperbolic ones with any positive weight function (Gao–Xie; building on Guilhot–Parkinson for $\tilde{B}_2,\tilde{G}_2$); and for complete-graph and right-angled Coxeter groups with any positive weight function (Xie). Lusztig's 2020 open-problems note still listed boundedness and the associativity of $J$ as open for general $(W,L)$. In 2025 Chen–Hu proved that every positively weighted finite-rank Coxeter group admits a bound, so the $a$-function is bounded in full generality; the sharp form of the boundedness conjecture (that $N_L(W)=\max_I L(w_I)$, over finite parabolic subgroups $W_I$ with longest element $w_I$, is a bound) remains open. What remains open after these advances is the heart of the original question: P1–P15 for general, non-quasisplit unequal positive weight functions, where neither Soergel–Hodge methods (equal parameters) nor perverse-sheaf geometry (crystallographic cases) apply.

Problem Statement

Let $(W,S)$ be a Coxeter system of finite rank and $L\colon W\to\mathbb{N}$ a positive weight function, with $H$ the associated Iwahori–Hecke algebra over $\mathbb{Z}[v^{\pm 1}]$, Kazhdan–Lusztig basis $\{c_w\}$, and structure constants $h_{x,y,z}\in\mathbb{Z}[v^{\pm 1}]$ defined by $c_xc_y=\sum_z h_{x,y,z}c_z$. Since every such $(W,S,L)$ is bounded (Chen–Hu 2025) — boundedness is a standing assumption here, not one of P1–P15 — the $a$-function and the leading coefficients $\gamma_{x,y,z^{-1}}$ are defined. Determine whether Lusztig's conjectures P1–P15 (arXiv:math/0208154, §14.2), which concern the $a$-function, the leading $\gamma$-coefficients, the two-sided cells, and the well-definedness of the asymptotic Hecke algebra $J$ with its unit, hold for every positively weighted finite-rank Coxeter system — the equal-parameter and quasisplit cases being settled, the open question is the general unequal-parameter case. A proof for all positively weighted finite-rank Coxeter systems, or an explicit Coxeter system with a positive weight function violating one of P1–P15, settles the question in its stated generality.

References

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