For a finite Coxeter group WW with parameter cc, Bonnafé and Rouquier conjectured that the Calogero–Moser cc-cells (defined through the ramification theory of the Calogero–Moser space; commuting Gaudin-type operators on RW\mathbb{R}W give only a partially conjectural description of left cells, and no operator-theoretic definition of two-sided cells is known) coincide with the Kazhdan–Lusztig cc-cells, together with their cellular characters and families. This was known for symmetric groups (type A) and Bonnafé and Germoni proved it for dihedral groups at equal parameters. The linked open problems left by that work are: (i) the strong spectral simplicity conjecture for the Gaudin family Dc,v,v′\mathscr{D}^{c,v,v'} for arbitrary parameters cc and arbitrary regular vectors (v,v′)(v,v'), beyond type A and beyond the one-ray case proved for dihedral groups; (ii) an intrinsic spectral or operator-theoretic definition of Calogero–Moser two-sided cells, removing the auxiliary choices of the current ramification-theoretic definition; and (iii) the coincidence of Calogero–Moser and Kazhdan–Lusztig cells, characters, and families for general finite Coxeter groups and unequal parameters. No resolution of any of these three problems is known in the literature.

Background

Let VV be a finite-dimensional real Euclidean space with inner product (.,.)(.,.) and let WW be a finite Coxeter group, that is, a finite group generated by orthogonal reflections of VV; its root system is denoted Φ\Phi, the positive roots by Φ+\Phi^+, its set of reflections by Ref(W)\mathrm{Ref}(W), and VregV_{\mathrm{reg}} is the open set of vectors with trivial stabilizer in WW. Fix v0=∑α∈Φ+αv_0=\sum_{\alpha\in\Phi^+}\alpha and let w0w_0 denote the longest element of WW, the unique element of WW sending every positive root to a negative root. A parameter is a WW-conjugacy-invariant real-valued function cc on Ref(W)\mathrm{Ref}(W); the space of parameters is denoted CR\mathscr{C}_{\mathbb{R}}, and cc is said to give equal parameters when cc is constant.

On one side stands Kazhdan–Lusztig theory. From the Hecke algebra of (W,c)(W,c) — a deformation of the group algebra of WW governed by the parameters cc — and its Kazhdan–Lusztig basis, the distinguished basis introduced by Kazhdan and Lusztig and characterized by certain polynomial positivity conditions, one obtains partitions of WW into left, right, and two-sided cc-cells. To each left cc-cell CC is attached a cellular character χCc,KL\chi^{c,KL}_C of WW, and the two-sided cc-cells partition the set Irr(W)\mathrm{Irr}(W) of irreducible characters into Kazhdan–Lusztig cc-families IrrΓc,KL(W)\mathrm{Irr}^{c,KL}_{\Gamma}(W). These partitions control the representation theory of Hecke algebras and, through Lusztig's classification of unipotent characters of finite reductive groups, are central to Lie-theoretic representation theory.

On the other side stands Calogero–Moser theory, defined by Rouquier and Bonnafé for any finite complex reflection group. The Calogero–Moser space is a symplectic variety obtained by Hamiltonian reduction — the standard symplectic-geometric construction that quotients a symplectic variety by a group action after fixing the values of the moment map, i.e. of the conserved quantities generating the action — from the rational Cherednik algebra at t=0t=0 (a deformation of the algebra of differential operators attached to WW and cc); its geometry encodes the representation theory of WW. Calogero–Moser (CM) left, right, and two-sided cc-cells were defined from the ramification theory of this space — how its projection to orbits degenerates — and this definition, especially for two-sided cells, depends on auxiliary algebraic choices such as a prime ideal in a Galois closure of a ring extension.

A bridge between the two theories is provided by Gaudin operators, defined for y∈Vy\in V, v∈Vregv\in V_{\mathrm{reg}}, v′∈Vv'\in V as endomorphisms of the group algebra RW\mathbb{R}W by

Dyc,v,v′(w)=(y,w−1(v′)) w−∑α∈Φ+csα (y,α)(v,α) w sα,\mathscr{D}^{c,v,v'}_y(w)=(y,w^{-1}(v'))\,w-\sum_{\alpha\in\Phi^+}c_{s_\alpha}\,\frac{(y,\alpha)}{(v,\alpha)}\,w\,s_\alpha,

where sαs_\alpha is the reflection with root α\alpha. These operators, analogous to the Dunkl operators of Cherednik algebra theory — the commuting deformations of directional derivatives built from the same reflection data — pairwise commute and are real symmetric, hence simultaneously diagonalizable; the family Dc,v,v′=(Dyc,v,v′)y∈V\mathscr{D}^{c,v,v'}=(\mathscr{D}^{c,v,v'}_y)_{y\in V} has simple spectrum when its joint eigenspaces in RW\mathbb{R}W are all at most one-dimensional, i.e. its spectrum Spc,v,v′\mathrm{Sp}^{c,v,v'} has cardinality ∣W∣|W|. Bonnafé and Rouquier showed that the joint eigenspaces of Dc,v0,0\mathscr{D}^{c,v_0,0} carry RW\mathbb{R}W-module structures whose characters are the CM cc-cellular characters (this family is not simple in general, so no simplicity hypothesis enters here), and that, under the ray-simplicity hypothesis (published Conjecture 4.2), CM left cells admit a (partially conjectural) definition by tracking joint eigenspaces of Dγ(t)c,v1,ξ(t)v2\mathscr{D}^{\gamma(t)c,v_1,\xi(t)v_2} along a path of parameters; the eigenvalue-to-cell incidence plays the role of the Kazhdan–Lusztig basis in this dictionary.

The known cases are: type A, where the strong spectral simplicity statement follows from work of Mukhin–Tarasov–Varchenko on the Bethe algebra of the Gaudin model — the commutative algebra generated by the Gaudin Hamiltonians, whose simultaneous eigenvectors are described by the Bethe ansatz — and where Brochier–Gordon–White proved the coincidence of CM and KL cells, cellular characters, and families using Gaudin algebras and the Robinson–Schensted–Knuth correspondence (the classical bijection between words and pairs of standard Young tableaux); and, proved by Bonnafé and Germoni, the dihedral groups W=I2(d)W=I_2(d), d>3d>3 — the symmetry groups of regular dd-gons, i.e. the rank-two real reflection groups — at equal parameters, where an explicit trigonometric diagonalization of the Gaudin operators in rank two establishes the weaker ray-wise simplicity statement (Conjecture 4.2 in the published numbering) and the coincidence of cells (Conjectures 5.4 and 5.7). The trivial case c=0c=0 is also elementary.

Problem Statement

The problem, as left open by Bonnafé–Germoni (arXiv:2112.13683; published as Conjectures 4.1, 5.4, 5.7 and §5.3 of Ann. Math. Blaise Pascal 29 (2023) 187–208) within the Bonnafé–Rouquier program, has one overarching objective: give Calogero–Moser cell theory an intrinsic spectral foundation and prove that it coincides with Kazhdan–Lusztig theory for finite Coxeter groups. The problem comprises three linked subgoals:

(1) Strong spectral simplicity: prove that for every finite Coxeter group WW, every parameter c∈CRc\in\mathscr{C}_{\mathbb{R}}, and every pair of regular vectors (v,v′)∈Vreg×Vreg(v,v')\in V_{\mathrm{reg}}\times V_{\mathrm{reg}}, the Gaudin family Dc,v,v′\mathscr{D}^{c,v,v'} has simple spectrum (published Conjecture 4.1). This is known for type A (Mukhin–Tarasov–Varchenko); for dihedral groups it is proved only in the ray case (ξv0,ξ′v0)(\xi v_0,\xi' v_0), ξ,ξ′∈R>0\xi,\xi'\in\mathbb{R}_{>0}, at constant cc (published Conjecture 4.2, a special case proved by Bonnafé–Germoni), and the extension to general (v,v′)(v,v'), to non-constant cc, and to other Coxeter types is open. Simplicity is the hypothesis under which the Gaudin-operator description of CM left cells is available, so subgoals (1) and (3) are linked. A proof or, equally, an explicit refutation of Conjecture 4.1 (an explicit finite Coxeter group WW, parameter cc, and regular vectors (v,v′)∈Vreg×Vreg(v,v')\in V_{\mathrm{reg}}\times V_{\mathrm{reg}} whose Gaudin family Dc,v,v′\mathscr{D}^{c,v,v'} has non-simple spectrum) resolves subgoal (1).

(2) Intrinsic two-sided cells: construct a definition of Calogero–Moser two-sided cc-cells, and of their associated cc-families IrrΓc,CM(W)\mathrm{Irr}^{c,CM}_{\Gamma}(W), in terms of intrinsic spectral or operator-theoretic data, removing the dependence of the current ramification-theoretic definition on auxiliary choices such as a prime ideal in a Galois closure; the definition must be proved well-defined and to reproduce the ramification-theoretic two-sided cells.

(3) The coincidence conjecture: prove, for finite Coxeter groups and parameters beyond the known cases — that is, beyond type A and beyond dihedral groups at equal parameters, including unequal (non-constant) cc and including non-constant cc for dihedral groups — that Calogero–Moser left (resp. two-sided) cc-cells coincide with Kazhdan–Lusztig left (resp. two-sided) cc-cells, that the CM and KL cellular characters of corresponding left cells are equal (published Conjecture 5.4), and that the CM and KL cc-families of corresponding two-sided cells are equal (published Conjecture 5.7). Within (3), Conjecture 5.4 (left-cell coincidence and equality of cellular characters) and Conjecture 5.7 (two-sided-cell coincidence and equality of families) are separately refutable sub-claims: a solution of either resolves that sub-claim and leaves the other open.

References

  1. Cédric Bonnafé and Jérôme Germoni, "Calogero–Moser cells of dihedral groups at equal parameters", Ann. Math. Blaise Pascal 29 (2023) 187–208, DOI 10.5802/ambp.410 (arXiv:2112.13683), https://doi.org/10.5802/ambp.410
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