Lusztig P1–P15 are not universally proved cell properties

In Lusztig's §13.1, preserving the setup of §3.1, one assumes $L(s)>0$ for every $s\in S$. The coefficient $f_{x,y,z}$ is defined by the standard-basis product $T_x{T}_y={\sum }_z{f}_{x,y,z}{T}_z$; $f_{x,y,z}^{\mathrm{\prime }}$ is defined by the same product in the $c$-basis, and $h_{x,y,z}$ by $c_x{c}_y={\sum }_z{h}_{x,y,z}{c}_z$. Thus the $f$-coefficients do not belong to an auxiliary $t$-basis. Section 13.2 calls $(W,L)$ bounded when there exists $N\in \mathbb{N}$ such that $v^{-N}{f}_{x,y,z}\in A_{\le 0}$ for all $x,y,z\in W$. Section 14.1 retains the positive-weight assumption and assumes boundedness; §14.2 labels P1–P15 as conjectures in that setup. Lusztig verifies them in specified cases in later sections, but the general list is not a universally established theorem.

Geck separately notes in Remark 5.9 of his Modular Representations of Hecke Algebras that the arguments of his §5 (including the result he attributes there to Lusztig §5.2) also go through for $L:W\to \mathbb{N}$ with some $L(s)=0$, provided $\mathrm{\Delta }(z)$ is set to $\mathrm{\infty }$ when the trace coefficient $\tau (c_z)$ vanishes. This is a qualification for those §5 arguments, not a removal of the positive-weight hypothesis from Lusztig's §§13–14.

The general basis and quotient constructions establish the weighted bar-invariant basis and the ideal and quotient-module construction for all stated Coxeter systems. Those constructions prove none of P1–P15, no positivity theorem, no semisimplicity theorem, and no classification of irreducible representations. A separate finite-Weyl theorem on this page proves modular canonical-basic-set triangularity under the explicitly stated P2–P8 and P15′ hypotheses and good-characteristic condition; it does not prove those hypotheses universally. Any stronger cell property must be stated with its exact type, parameter, boundedness, and proof hypotheses.