Characterising Kazhdan–Lusztig left cells by two-sided cells and a generalised tau-invariant with unequal parameters (Bonnafé–Geck, Conjecture 7.5)

For a Coxeter group $W$ with an arbitrary positive weight function, Bonnafé and Geck (2015) introduced a generalised $\tau$-invariant built from strongly admissible operations on dihedral parabolic subgroups together with an enhanced right descent set, and conjectured that two elements of $W$ lie in the same left Kazhdan–Lusztig cell exactly when they lie in the same two-sided cell and share this invariant. The forward implication is a theorem, and the conjecture has been verified computationally in small rank (types $F_4$ and $B_n$ for $n\leqslant 7$ with all weight functions, and, in a companion equal-parameter formulation, all exceptional types and $B_n$, $D_n$ for $n\leqslant 9$); for type $B_n$ with asymptotic and intermediate weights it is known that left cells coincide with Vogan classes for suitably enlarged collections of admissible operations. The general statement, including the equal-parameter case for unbounded-rank types $B_n$ and $D_n$, remains open. The problem is to prove or disprove the conjecture for arbitrary Coxeter systems and positive weight functions.

Background

Let $(W,S)$ be a Coxeter system, that is, a group $W$ generated by a finite set $S$ of involutions subject only to the relations $(st)^{m(s,t)}=1$ for integers $m(s,t)\geqslant 2$. A weight function is a map $L:W\to\mathbb{Z}$ with $L(vw)=L(v)+L(w)$ whenever the lengths add; it is determined by positive integers $p_s=L(s)$ on $S$, and the case $p_s$ all equal is the classical equal-parameter case. To a weighted Coxeter system one associates the Iwahori–Hecke algebra $\mathcal{H}$, a deformation of the group algebra of $W$ with one parameter $v^{p_s}$ per simple reflection, and Kazhdan and Lusztig showed in 1979 that the Kazhdan–Lusztig basis of $\mathcal{H}$ induces preorders $\leqslant_L$, $\leqslant_R$ and $\leqslant_{LR}$ on $W$, whose equivalence classes are the left, right and two-sided cells. Cells with unequal parameters govern the representation theory of reductive groups over $p$-adic fields and are systematically developed in Lusztig's monograph on Hecke algebras with unequal parameters; determining the cell partitions for general weights is a basic and largely open problem.

A classical tool for describing left cells is Vogan's $\tau$-invariant (1979), introduced to classify primitive ideals in enveloping algebras and transported to Coxeter groups by a standard dictionary. Its starting point is the right descent set $\mathcal{R}(w)=\{s\in S: \ell(ws)<\ell(w)\}$ (where $\ell$ is the length function), which is constant on left cells; the full generalised $\tau$-invariant also records how these descent sets behave under the Kazhdan–Lusztig star operations, which act along rank-two (dihedral) parabolic subgroups $W_{\{s,t\}}$. In type $A_n$ this invariant is complete within two-sided cells, which underlies the description of cells via the Robinson–Schensted correspondence — the classical bijection between permutations and pairs of standard Young tableaux — found by Kazhdan and Lusztig, but in general the invariant is not complete.

Bonnafé and Geck (2015; preprint version Geck 2014) proposed a framework that extends Vogan's construction to arbitrary weights. A pair $(I,\delta)$ consisting of a subset $I\subseteq S$ and a map $\delta:W_I\to W_I$ on the parabolic subgroup $W_I$ is called (strongly) admissible if $\delta$ permutes the left cells of $W_I$, induces isomorphisms of the corresponding cell representations of the parabolic Hecke algebra (and preserves right cells); each such $\delta$ has a canonical extension $\delta^L$ to all of $W$. For dihedral $I=\{s,t\}$ with $s\neq t$ there are two key strongly admissible involutions: $\delta_{s,t}$, available when $p_s=p_t$ and $m(s,t)\geqslant 3$, and $\delta^{<}_{s,t}$, available when $p_s<p_t$ and $m(s,t)$ is even with $m(s,t)\geqslant 4$ (both defined below). Writing $S_p=S\cup\{sts: s,t\in S,\ p_s<p_t\}$, the enhanced right descent set is $\mathcal{R}_p(w)=\{s\in S_p: \ell(ws)<\ell(w)\}$, again constant on left cells. The interplay of the maps $\delta^L$ with $\mathcal{R}_p$ produces, by a finite induction reviewed below, the generalised $\tau^{\Delta_2,\mathcal{R}_p}$-invariant whose fibres are called left Vogan $(\Delta_2,\mathcal{R}_p)$-classes; here $\Delta_2$ is the collection of the dihedral pairs $(I,\delta)$ just described. Bonnafé (2014) developed the parallel purely group-theoretic notion of Vogan classes and proved that, with unequal parameters as in the classical equal-parameter case, Vogan classes are unions of left cells; Bonnafé and Geck proved the analogous forward implication (same left cell implies same two-sided cell and same generalised $\tau$-invariant) in full generality. The two-sided-cell ingredient is a genuine restriction: by work of Lusztig recalled in the chapter, two-sided cells are captured by Lusztig's $\mathbf{a}$-function, an integer-valued function on $W$ defined from the Hecke algebra that is constant on two-sided cells, so the conjecture asserts that within a two-sided cell the purely local dihedral data of $\mathcal{R}_p$ and the $\delta^L$ completely cut out the left cells.

Evidence: in the equal-parameter case the companion statement 'same left cell if and only if same $\mathbf{a}$-value and same $\tilde\tau$-cell' (Geck–Halls, Conjecture 6.9) was verified by computation for all exceptional types, including the non-crystallographic types $H_3$ and $H_4$ (types whose Coxeter diagram does not arise from a lattice of roots, hence which are not Weyl groups), and for types $B_n$, $D_n$ with $n\leqslant 9$, while the general large-rank cases $B_n$, $D_n$ remained unproved; with unequal parameters the new conjecture was checked for $F_4$ and $B_n$ ($n\leqslant 7$) for all weight functions using Geck's PyCox package. For type $B_n$ with weights close to asymptotic, Howse (2019) and Howse–Pietraho (2019), building on the earlier description of asymptotic cells by Bonnafé and Iancu and the conjectures of Bonnafé–Geck–Iancu–Lam, proved that left cells do coincide with Vogan classes for suitably enlarged collections of admissible operations when $b/a\geqslant n-1$. On the other hand, the classification of primitive ideals for complex classical Lie algebras was completed in type D only by McGovern and Pietraho (2023), who had to enlarge the generalised $\tau$-invariant with operators attached to rank-four ($D_4$-type) parabolics rather than dihedral ones — a warning that dihedral-level data may not always suffice in type D, precisely one of the types left open in the equal-parameter case.

Problem Statement

Let $(W,S)$ be a Coxeter system with a weight function $L$ given by positive integers $p_s=L(s)$, and let cells be the Kazhdan–Lusztig cells of the corresponding Iwahori–Hecke algebra with parameters $(v^{p_s})_{s\in S}$. Define, following Bonnafé–Geck (2015):

1. $S_p=S\cup\{sts:\ s,t\in S,\ s\neq t,\ p_s<p_t\}$ and, for $w\in W$, the enhanced right descent set $\mathcal{R}_p(w)=\{s\in S_p:\ \ell(ws)<\ell(w)\}$, where $\ell$ is the length function.
2. For $I=\{s,t\}\subseteq S$ with $s\neq t$ and $m=m(s,t)<\infty$, write the elements of the dihedral group $W_I$ in normal form $1_k=sts\cdots$ and $2_k=tst\cdots$ ($k$ factors), and let $w_{s,t}$ denote the longest element of $W_I$. If $p_s=p_t$ and $m\geqslant 3$, let $\delta_{s,t}:W_I\to W_I$ be the involution $\delta(1_0)=1_0$, $\delta(2_m)=2_m$, $\delta(1_k)=1_{m-k}$, $\delta(2_k)=2_{m-k}$ for $1\leqslant k\leqslant m-1$. If $p_s<p_t$ and $m$ is even with $m\geqslant 4$, let $\delta^{<}_{s,t}:W_I\to W_I$ be the involution that fixes $1$, $s$, $w_{s,t}$ and $w_{s,t}s$ and sends every other element $w\in W_I$ to $ws$; in normal forms, it fixes $1_0$, $1_1$, $2_{m-1}$, $2_m$ and interchanges $1_{2i}\leftrightarrow 1_{2i+1}$ for $1\leqslant 2i<2i+1\leqslant m-1$ and $2_{2i+1}\leftrightarrow 2_{2i+2}$ for $1\leqslant 2i+1<2i+2\leqslant m-2$. Both involutions are strongly admissible, and each extends canonically to $\delta^L:W\to W$ by $\delta^L(xu)=x\,\delta(u)$ where $w=xu$ with $x$ the minimal coset representative of $w$ modulo $W_I$ and $u\in W_I$.
3. Let $\Delta_2$ be the collection of all pairs $(I,\delta)$ with $I=\{s,t\}$, $s\neq t$, taken with $p_s\leqslant p_t$, where $\delta=\delta_{s,t}$ when $p_s=p_t$ and $m(s,t)\geqslant 3$, and $\delta=\delta^{<}_{s,t}$ when $p_s<p_t$ and $m(s,t)$ is even with $m(s,t)\geqslant 4$. For $y,w\in W$ define inductively $y\approx_0 w$ if $\mathcal{R}_p(y)=\mathcal{R}_p(w)$, and $y\approx_n w$ if $y\approx_{n-1}w$ and $\delta^L(y)\approx_{n-1}\delta^L(w)$ for every $(I,\delta)\in\Delta_2$; write $y\approx^{\Delta_2,\mathcal{R}_p}w$ if $y\approx_n w$ for all $n\geqslant 0$. By Bonnafé–Geck (Remark 7.3) this is equivalent to their Definition 7.1 of the generalised $\tau^{\Delta_2,\mathcal{R}_p}$-invariant, whose fibres are the left Vogan $(\Delta_2,\mathcal{R}_p)$-classes.

Conjecture (Bonnafé–Geck, Conjecture 7.5; preprint version Geck, Conjecture 4.9). For all $x,y\in W$:

$x\sim_L y \quad\text{if and only if}\quad x\sim_{LR}y \ \text{ and }\ x\approx^{\Delta_2,\mathcal{R}_p}y,$

where $\sim_L$ and $\sim_{LR}$ denote membership in the same left cell and the same two-sided cell. Prove this statement, or disprove it by an explicit counterexample, for arbitrary Coxeter systems and positive weight functions. The forward implication is already known and may be cited; the problem is the converse, together with the equal-parameter cases of types $B_n$ and $D_n$ of unbounded rank. Any answer must respect the original generality: partial results restricted to particular types or weight regimes do not constitute a resolution.

References

1. Cédric Bonnafé and Meinolf Geck, 'Hecke algebras with unequal parameters and Vogan's left cell invariants', in: Representations of Reductive Groups, Progress in Mathematics 312, Birkhäuser/Springer, Cham, 2015, pp. 173–187, DOI 10.1007/978-3-319-23443-4_6, https://doi.org/10.1007/978-3-319-23443-4_6 (primary published source of the conjecture, stated there as Conjecture 7.5; open-access copy at https://hal.science/hal-01113388)
2. Meinolf Geck, 'A generalised $\tau$-invariant for the unequal parameter case', preprint (2014), arXiv:1405.5736, DOI 10.48550/arXiv.1405.5736, https://arxiv.org/abs/1405.5736 (single-author preprint version of the same work; the conjecture is numbered Conjecture 4.9 there)
3. Meinolf Geck and Abbie Halls, 'On the Kazhdan–Lusztig cells in type $E_8$', Mathematics of Computation 84 (2015), no. 296, 3029–3049, DOI 10.1090/mcom/2963, https://doi.org/10.1090/mcom/2963 (arXiv:1401.6804; contains the equal-parameter companion Conjecture 6.9 and its computational verification)
4. Cédric Bonnafé, 'Vogan classes and cells in the unequal parameter case', preprint (2014), arXiv:1405.6470, DOI 10.48550/arXiv.1405.6470, https://arxiv.org/abs/1405.6470 (absorbed as the chapter 'Descent Sets, Knuth Relations and Vogan Classes', pp. 133–137, in C. Bonnafé, Kazhdan–Lusztig Cells with Unequal Parameters, Algebra and Applications 24, Springer, 2017, DOI 10.1007/978-3-319-70736-5_9)
5. Edmund Howse, 'Vogan classes in type $B_n$', Algebraic Combinatorics 2 (2019), no. 6, 1033–1057, DOI 10.5802/alco.74, https://doi.org/10.5802/alco.74 (arXiv:1709.02061)
6. Edmund Howse and Thomas Pietraho, 'Kazhdan–Lusztig left cells in type $B_n$ for intermediate parameters', preprint (2019), arXiv:1902.09301, DOI 10.48550/arXiv.1902.09301, https://arxiv.org/abs/1902.09301
7. William McGovern and Thomas Pietraho, 'On the classification of primitive ideals for complex classical Lie algebras, IV', preprint (2023, revised 2026), arXiv:2309.02363, DOI 10.48550/arXiv.2309.02363, https://arxiv.org/abs/2309.02363
8. David Kazhdan and 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
9. D. A. Vogan, Jr., 'A generalized $\tau$-invariant for the primitive spectrum of a semisimple Lie algebra', Mathematische Annalen 242 (1979), 209–224, DOI 10.1007/BF01420727, https://doi.org/10.1007/BF01420727
10. G. Lusztig, 'Hecke Algebras with Unequal Parameters', CRM Monograph Series 18, American Mathematical Society, Providence, RI, 2003, DOI 10.1090/crmm/018, ISBN 978-0-8218-3356-8, https://doi.org/10.1090/crmm/018
11. Cédric Bonnafé and Lacrimioara Iancu, 'Left cells in type $B_n$ with unequal parameters', Representation Theory 7 (2003), 587–609, DOI 10.1090/S1088-4165-03-00188-2, https://doi.org/10.1090/S1088-4165-03-00188-2
12. Cédric Bonnafé, Meinolf Geck, Lacrimioara Iancu and 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/Springer, 2010, pp. 33–54, DOI 10.1007/978-0-8176-4697-4_3, https://doi.org/10.1007/978-0-8176-4697-4_3
13. Meinolf Geck, 'PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras', LMS Journal of Computation and Mathematics 15 (2012), 231–256, DOI 10.1112/s1461157012001064, https://doi.org/10.1112/s1461157012001064