The Bonnafé–Rouquier conjecture: equality of Calogero–Moser cells and Kazhdan–Lusztig cells for finite Coxeter groups beyond type A and rank two

The Bonnafé–Rouquier conjecture predicts that, for every finite Coxeter group $W$ and matching parameter, the Calogero–Moser cells — partitions of $W$ defined through the ramification theory of the Calogero–Moser space of the rational Cherednik algebra of $W$ at $t=0$ — coincide with Lusztig's Kazhdan–Lusztig cells. Brochier–Gordon–White (2020) confirmed the conjecture for symmetric groups by combining inhomogeneous Gaudin algebras, the RSK correspondence and crystal bases, and the Mukhin–Tarasov–Varchenko realization of the Calogero–Moser phase space inside the Gaudin spectrum. The conjecture is still open beyond type A, the dihedral groups at equal parameter, and type $B_2$: the imprimitive Coxeter types of rank at least 3, type $D$, the exceptional Weyl groups, the non-crystallographic groups $H_3$ and $H_4$, and non-constant dihedral parameters beyond the established cases remain unresolved, and the extension of the cell framework to non-real complex reflection groups is a further open program.

Background

In 1979 Kazhdan and Lusztig introduced, for a finite Coxeter group $(W,S)$ — that is, a finite group generated by reflections of a real Euclidean space, for example a Weyl group — a partition of the elements of $W$ into left, right, and two-sided cells. These cells are defined through the Kazhdan–Lusztig basis of the Hecke algebra, a one-parameter deformation of the group algebra of $W$, and they organize much of the representation theory of $W$ and of its Hecke algebra: cell modules yield representations, and two-sided cells partition the irreducible characters into Lusztig families. Lusztig later extended the theory to Hecke algebras with unequal parameters, where the cells depend on a weight function on the reflections.

A complex reflection group is a finite subgroup of $GL(V)$ generated by pseudo-reflections, i.e. elements fixing a hyperplane pointwise; not every such group is a Coxeter group. To every complex reflection group $W$ one attaches the rational Cherednik algebra $H_{t,c}(W)$, a two-parameter family of associative algebras deforming the natural action of $W$ on the symmetric algebra of $V\oplus V^{*}$; the parameters are a number $t$ and a function $c$ on the reflections, constant on conjugacy classes. At $t=0$ the algebra $H_{0,c}(W)$ has a large commutative center, and the spectrum of that center is the Calogero–Moser space, an affine symplectic variety deforming the quotient $(V\times V^{*})/W$. Bonnafé and Rouquier studied the Galois theory of the covering of smooth loci from the Calogero–Moser space to $V/W\times V^{*}/W$ and used its ramification — the inertia groups over points of the base — to define, for every parameter $c$, partitions of $W$ into left, right, and two-sided Calogero–Moser cells, together with Calogero–Moser families of irreducible characters and Calogero–Moser cellular representations. Their conjecture, stated in 2013 and developed in a book-length manuscript, predicts that when $W$ is a Coxeter group and the Cherednik parameter $c$ matches a Lusztig weight function (the constant $c$ case corresponding to the original equal-parameter Kazhdan–Lusztig theory), the Calogero–Moser partitions coincide with the Kazhdan–Lusztig left, right, and two-sided cells, with the families and cellular representations matching accordingly. Conceptually, the conjecture says that Kazhdan–Lusztig cell theory is the Hecke-algebra shadow of the geometry of Calogero–Moser spaces, and that this geometry provides the correct notion of cells for all complex reflection groups, where no Hecke-theoretic cell theory is available.

Brochier, Gordon, and White confirmed the conjecture for the symmetric groups $W=\mathfrak{S}_n$. Their route is special to type A: the inhomogeneous Gaudin algebras — maximal commutative subalgebras of $U(\mathfrak{gl}_r)^{\otimes n}$ deforming the Hamiltonians of the inhomogeneous Gaudin quantum spin chain — act on tensor products of polynomial representations; work of Halacheva–Kamnitzer–Rybnikov–Weekes shows that the monodromy of the joint spectrum along degeneration paths is described by the cactus group — the group whose generators perform partial reversals of tensor factors, governing how tensor products of representations degenerate — acting on crystal bases (combinatorial, tableaux-shaped directed graphs that encode the structure of quantum-group representations) through the Robinson–Schensted–Knuth (RSK) correspondence, the classical bijection between words and pairs of standard tableaux; and a theorem of Mukhin–Tarasov–Varchenko realizes the rational Calogero–Moser phase space as a part of this Gaudin spectrum. Combining these inputs, the authors computed the Calogero–Moser cells of $\mathfrak{S}_n$ and matched them with the known Kazhdan–Lusztig cells. None of these tools extends beyond type A, which is why the general conjecture remains the open target.

Problem Statement

Prove, or disprove by an explicit counterexample, the Bonnafé–Rouquier cell conjecture: for every finite Coxeter group $W$ and every Cherednik parameter $c$ on $W$ matching a Lusztig weight function — in the sense of the original Bonnafé–Rouquier conjecture (References 2 and 3), which fixes the correspondence between the parameter $c$ and Lusztig weight functions; in particular this includes the equal-parameter case — the Calogero–Moser left, right, and two-sided cells of $W$ — defined via the ramification of the Calogero–Moser covering arising from the center of the rational Cherednik algebra $H_{0,c}(W)$ — coincide with Lusztig's Kazhdan–Lusztig left, right, and two-sided cells for the corresponding Hecke-algebra parameters, together with the accompanying identifications of Calogero–Moser two-sided-cell families with Lusztig families of irreducible characters and of Calogero–Moser cellular representations with Kazhdan–Lusztig cell representations. The statement is already established for the type-A Coxeter groups (symmetric groups, all ranks), for the dihedral groups $I_2(m)$ at equal parameter, and for type $B_2$ as recorded in the Bonnafé–Rouquier manuscript; it remains open for the imprimitive types $B_n/C_n$ with $n\ge 3$ and $D_n$ with $n\ge 4$, the exceptional Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, the non-crystallographic groups $H_3$ and $H_4$, and the dihedral groups with non-constant parameter not covered by those established results, for all parameter values $c$ matching a Lusztig weight function.

References

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