Semicontinuity of Kazhdan–Lusztig cells with unequal parameters: Bonnafé's Conjectures A, A′, A″ and B

For a Coxeter system $(W,S)$ with $S$ finite, the partition of $W$ into Kazhdan–Lusztig left, right and two-sided cells depends on the choice of weight function on $W$. Bonnafé (arXiv:0808.3522; New Zealand J. Math. 39 (2009) 171–192) conjectured a precise semicontinuity law for this dependence: there should exist a finite complete rational hyperplane arrangement in the cone of positive subsets of $\mathbb{Z}[\bar{S}]$ (equivalently, a finite set of essential integer linear forms) such that cell partitions are constant on arrangement facets, with an explicit 'join' description at the walls (Conjecture A, in equivalent forms A′ and A″), and this degeneration should be compatible with cell representations through filtrations of limit cell modules (Conjecture B) and character additivity (Conjecture B⁻). Clause (a) is known for all finite $W$ (Geck), and full cases are verified only for dihedral groups, type $F_4$, and affine rank-2 groups $\tilde{G}_2$, $\tilde{B}_2$, while for type $B_n$ the predicted critical ratios $1,2,\dots,n-1$ are confirmed by computation up to $n\le6$. Clause (b) of Conjecture A and Conjectures B/B⁻ remain open in general — clause (b) open even for finite $W$ — with the conjectures still stated as conjectures in Bonnafé's 2017 monograph.

Background

A Coxeter system $(W,S)$ is a group $W$ generated by a set $S$ of involutions subject only to relations fixing the order of each product $st$; the prototypical examples are finite Weyl groups (symmetry groups of root systems) and affine Weyl groups (infinite crystallographic reflection groups). To such a system one attaches the Iwahori–Hecke algebra $\mathcal{H}(W)$, a deformation of the group algebra of $W$ with one parameter per conjugacy class in $S$; the 'unequal parameters' setting means these parameters may differ. Formally one fixes a weight function $L:W\to\mathbb{Z}$ with $L(ww')=L(w)+L(w')$ whenever $\ell(ww')=\ell(w)+\ell(w')$ ($\ell$ the length function), so $L$ is determined by its values on $S$, constant on conjugacy classes; writing $\bar{S}$ for the set of $W$-conjugacy classes of simple reflections, a weight function is essentially a linear form on the free abelian group $\mathbb{Z}[\bar{S}]$. Kazhdan and Lusztig (1979) constructed a distinguished basis $(c_w)_{w\in W}$ of the Hecke algebra (the Kazhdan–Lusztig basis), defined using polynomials that compute intersection cohomology of Schubert varieties in the equal-parameter case; the multiplication rules $c_xc_y=\sum_z \mu(x,y,z)c_z$ induce preorders on $W$ whose equivalence classes are the left, right, and two-sided Kazhdan–Lusztig cells. Cells control the representation theory: each left cell $C$ carries a cell representation $\mathbb{Z}M_C$, and in Lusztig's theory of families of characters for finite groups of Lie type the two-sided cells for unequal parameters govern unipotent character families. Lusztig's book 'Hecke algebras with unequal parameters' (2003) developed the general unequal-parameter theory and stated a framework of conjectures (P1–P15) about cells for a fixed weight function, including the a-function, an integer-valued invariant expected to be constant on two-sided cells and finite-valued. A separate strand, originating in Rouquier's work and developed by Chlouveraki, showed that for finite complex reflection groups certain block invariants of cyclotomic Hecke algebras depend on the parameters only through finitely many rational 'essential hyperplanes' in weight space. The source paper (Bonnafé, arXiv:0808.3522, published as New Zealand J. Math. 39 (2009), 171–192) transfers and sharpens this intuition for Kazhdan–Lusztig cells themselves. Since a weight function may be composed with any order-preserving embedding of its value group, the intrinsic parameter space is not the linear forms themselves but their 'positive subsets': a positive subset of $\mathbb{Z}[\bar{S}]$ is a subset $X$ with $\mathbb{Z}[\bar{S}]=X\cup(-X)$, $X+X\subseteq X$, and $X\cap(-X)$ a subgroup; the set $\mathrm{Pos}(\mathbb{Z}[\bar{S}])$ of these carries a natural topology, and every homomorphism $\varphi$ from $\mathbb{Z}[\bar{S}]$ to a totally ordered abelian group yields a positive subset $\mathrm{Pos}(\varphi)=\{\lambda\mid\varphi(\lambda)\ge0\}$ (this framework was set up in Bonnafé's companion note arXiv:0808.3518). A rational hyperplane arrangement $\mathfrak{A}$ in this space is a finite set of kernels of non-zero integer linear forms; its facets are the regions where all the defining linear forms have constant signs, and it is complete when it includes the coordinate hyperplanes. Computations of Geck for type $F_4$ (published in J. Algebra 281 (2004)) and of Guilhot for the affine Weyl groups $\tilde{G}_2$ and $\tilde{B}_2$, together with Bonnafé–Geck–Iancu–Lam's domino-insertion analysis of type $B_n$ (where CHEVIE computations up to $n\le6$ support the critical ratios $1,2,\dots,n-1$, giving Conjecture C below), suggested a universal law that the paper formalizes as Conjectures A, A′, A″ (about cells) and B, B⁻ (about cell representations).

Problem Statement

The problem is to settle Bonnafé's semicontinuity conjectures for Kazhdan–Lusztig cells with unequal parameters, stated for an arbitrary Coxeter system $(W,S)$ with $S$ finite and $\bar{S}$ the set of $W$-conjugacy classes of simple reflections. Let $\mathrm{Pos}(\mathbb{Z}[\bar{S}])$ denote the positive subsets of $\mathbb{Z}[\bar{S}]$; for $X\in\mathrm{Pos}(\mathbb{Z}[\bar{S}])$ and $?\in\{L,R,LR\}$ let $\sim^X_?$ be the corresponding left, right, or two-sided cell equivalence relation on $W$, and for a facet $F$ of a hyperplane arrangement let $W_F$ be the parabolic subgroup generated by the classes of simple reflections whose defining forms vanish on $F$.

Conjecture A. There exists a finite complete rational hyperplane arrangement $\mathfrak{A}$ in $\mathrm{Pos}(\mathbb{Z}[\bar{S}])$ such that: (a) if $X$ and $Y$ belong to the same $\mathfrak{A}$-facet $F$, then $\sim^X_?=\sim^Y_?$ (denoted $\sim^F_?$); (b) for each $\mathfrak{A}$-facet $F$, the cells for $\sim^F_?$ are the minimal subsets $C\subseteq W$ such that (b1) for every chamber $\mathcal{C}$ with $F\subseteq\overline{\mathcal{C}}$, $C$ is a union of cells for $\sim^{\mathcal{C}}_?$, and (b2) $C$ is stable under translation by $W_F$ (on the left, right, or both sides according as $?=L,R,LR$); equivalently $\sim^F_?=\sup(\mathrm{trans}^{W_F}_?,\sup_{F\subseteq\overline{\mathcal{C}}}\sim^{\mathcal{C}}_?)$, the join being generated as an equivalence relation.

When Conjecture A holds there is a unique minimal such arrangement (the essential hyperplanes of $(W,S)$), and the conjecture is equivalent to: Conjecture A′, there is a finite set $\mathcal{E}\subseteq\mathbb{Z}[\bar{S}]\setminus\{0\}$ containing $\bar{S}$ such that the cell relations for maps $\varphi:\bar{S}\to\Gamma$ into arbitrary totally ordered abelian groups depend only on the sign pattern $\mathrm{sgn}_\mathcal{E}(\varphi)\in\{+,-,0\}^{\mathcal{E}}$, with the wall description (b) as above; and, when $|\bar{S}|=2$, to: Conjecture A″, there are rational critical ratios $0<r_1<\dots<r_m$ (with $r_0=0$, $r_{m+1}=\infty$) such that the cell partition is constant in each interval $(r_i,r_{i+1})$, the cells at the wall $r_i$ are obtained by joining the cells of the two adjacent intervals, and the cells at the degenerate walls $0$ and $\infty$ (where one parameter vanishes) are obtained by joining the adjacent chamber cells with translations by the corresponding parabolic subgroup.

Conjecture B. Assuming Conjecture A, if $\mathcal{C}$ is an $\mathfrak{A}$-chamber with $X\in\overline{\mathcal{C}}$ and $Y\in\mathcal{C}$, so that a cell $C$ for $\sim^X_?$ decomposes as a disjoint union of cells $C_i$ for $\sim^{Y}_?$, then the cell module $\mathbb{Z}M^{?,X}_C$ admits a finite filtration whose graded pieces are direct sums of the modules $\mathbb{Z}M^{?,Y}_{C_i}$, with the index partition compatible with the cell order; and in the weaker form Conjecture B⁻ (for finite $W$), the character of the limit cell module is the sum of the characters of the adjacent-chamber cell modules: $\chi^{?,X}_C=\sum_i\chi^{?,Y}_{C_i}$.

The common objective unifying these conjectures is to establish precisely how the Kazhdan–Lusztig cell partition and its cell representations of any Coxeter system degenerate as the weight function varies: the facet-constancy statement (a), the explicit wall-join description (b), the equivalent essential-elements/critical-ratio formulations, the representation-theoretic compatibility (B/B⁻), and, as part of the same objective, the determination of the essential hyperplanes in the flagship open family of type $B_n$ (Conjecture C: the Conjectures A and B should hold for the Coxeter system $(W_n,S_n)$ of type $B_n$, and the essential hyperplanes should be exactly $H_s$, $H_t$, $H_{t\pm is}$ for $1\le i\le n-1$ (where $H_\lambda$ is the kernel of the linear form $\lambda$, and $\bar{S}_n=\{s,t\}$ with $t$ the long simple reflection and $s$ the class of the $n-1$ short simple reflections); the critical-ratio prediction $r_i=i$ is confirmed by computation up to $n\le6$). Note clause (a) is already a theorem for all finite $W$ (Geck), so the open heart of Conjecture A is clause (b) — open even for finite $W$ — together with the case of infinite $W$ generally.

The open parts are the following: the open part of Conjecture A, namely clause (b) for any $(W,S)$ with $S$ finite together with the whole conjecture for infinite $W$; Conjecture B, of which Conjecture B⁻ is the weaker finite-$W$ form; and the exact list of essential hyperplanes for type $B_n$, which is the non-redundant content of Conjecture C (its type-$B_n$ clauses for Conjectures A and B follow from the other two parts). Refuting any single part settles that part, and the equivalent restatements A′ and A″ are equivalent to Conjecture A.

References

1. Cédric Bonnafé, Semicontinuity properties of Kazhdan-Lusztig cells, arXiv:0808.3522 (2008, v2 2010), DOI 10.48550/arXiv.0808.3522, https://arxiv.org/abs/0808.3522; published version: New Zealand Journal of Mathematics 39 (2009), 171-192, https://zbmath.org/1231.20004
2. David Kazhdan, George Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones Mathematicae 53 (1979), 165-184, DOI 10.1007/bf01390031, https://doi.org/10.1007/bf01390031
3. George Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series 18, American Mathematical Society, Providence, RI, 2003, DOI 10.1090/crmm/018, https://doi.org/10.1090/crmm/018 (Zbl 1051.20003)
4. Meinolf Geck, Computing Kazhdan-Lusztig cells for unequal parameters, Journal of Algebra 281 (2004), 342-365, DOI 10.1016/j.jalgebra.2004.07.029, https://doi.org/10.1016/j.jalgebra.2004.07.029 (addendum: Journal of Algebra 292 (2005), 2, DOI 10.1016/j.jalgebra.2005.08.024)
5. Maria Chlouveraki, On the cyclotomic Hecke algebras of complex reflection groups, arXiv:0710.0776 (2007), DOI 10.48550/arXiv.0710.0776, https://arxiv.org/abs/0710.0776
6. Cédric Bonnafé, Topologie sur l'ensemble des parties positives d'un réseau, arXiv:0808.3518 (2008), DOI 10.48550/arXiv.0808.3518, https://arxiv.org/abs/0808.3518
7. Cédric Bonnafé, Meinolf Geck, Lacrimioara Iancu, Thomas Lam, On domino insertion and Kazhdan-Lusztig cells in type B_n, in: Representation Theory of Algebraic Groups and Quantum Groups (Progress in Mathematics 284), Birkhäuser, 2010, 33-54, DOI 10.1007/978-0-8176-4697-4_3, https://doi.org/10.1007/978-0-8176-4697-4_3 (Zbl 5919678)
8. Cédric Bonnafé, On Kazhdan-Lusztig cells in type B, Journal of Algebraic Combinatorics 31 (2010), no. 1, 53-82, DOI 10.1007/s10801-009-0183-2, https://doi.org/10.1007/s10801-009-0183-2 (erratum: Journal of Algebraic Combinatorics 35 (2012), 515-517, DOI 10.1007/s10801-012-0351-7; arXiv:0806.0214; Zbl 5682323)
9. Jérémie Guilhot, Kazhdan-Lusztig cells in affine Weyl groups of rank 2, International Mathematics Research Notices 2010 (17), 3422-3462, DOI 10.1093/imrn/rnp243, https://doi.org/10.1093/imrn/rnp243 (Zbl 5798865)
10. Jian-yi Shi, A counter-example to a conjecture of Lusztig, Journal of Algebra 323 (2010), 2591-2598, DOI 10.1016/j.jalgebra.2010.01.018, https://doi.org/10.1016/j.jalgebra.2010.01.018 (Zbl 1197.20033)
11. Cédric Bonnafé, Raphaël Rouquier, Calogero-Moser versus Kazhdan-Lusztig cells, Pacific Journal of Mathematics 261 (2013), 45-51, DOI 10.2140/pjm.2013.261.45, https://doi.org/10.2140/pjm.2013.261.45
12. Cédric Bonnafé, Meinolf Geck, Hecke algebras with unequal parameters and Vogan's left cell invariants, in: Representations of Reductive Groups (Progress in Mathematics 312), Birkhäuser/Springer, 2015, DOI 10.1007/978-3-319-23443-4_6, https://doi.org/10.1007/978-3-319-23443-4_6 (arXiv:1502.01661)
13. Cédric Bonnafé, Kazhdan-Lusztig cells with unequal parameters, Advanced Courses in Mathematics - CRM Barcelona, Springer, Cham, 2017, DOI 10.1007/978-3-319-70736-5, https://link.springer.com/book/10.1007/978-3-319-70736-5 (Semicontinuity chapter: DOI 10.1007/978-3-319-70736-5_7; Zbl 6856758)
14. Jérémie Guilhot, James Parkinson, A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters, Proceedings of the London Mathematical Society 117 (2018), 879-940, DOI 10.1112/plms.12211, https://doi.org/10.1112/plms.12211 (arXiv:1711.06551)
15. Jérémie Guilhot, James Parkinson, Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for $\tilde{C}_2$, Algebraic Combinatorics 2 (2019), 969-1031, DOI 10.5802/alco.75, https://doi.org/10.5802/alco.75 (arXiv:1803.11067)
16. Edmund Howse, Vogan classes in type $B_n$, Algebraic Combinatorics 2 (2019), 1033-1057, DOI 10.5802/alco.74, https://doi.org/10.5802/alco.74 (arXiv:1709.02061)
17. Jianwei Gao, Xun Xie, Conjectures P1-P15 for hyperbolic Coxeter groups of rank 3, Journal of Algebra 564 (2021), 125-152, DOI 10.1016/j.jalgebra.2020.09.020, https://doi.org/10.1016/j.jalgebra.2020.09.020 (arXiv:1906.09451)