Calogero–Moser Pairs in Integrable Systems

• Calogero–Moser pairs are defined as matrix pairs (X, Y) satisfying [X, Y] + I has rank 1, forming the basis for moduli spaces in algebraic geometry.
• They classify ideals in the Weyl algebra and serve as moduli points in Calogero–Moser spaces, linking noncommutative geometry with bispectral analysis.
• Their parametrization underlies classical and quantum integrable systems, with applications in reflection groups, KP hierarchies, and orthogonal polynomials.

A Calogero–Moser (CM) pair, in the context of integrable systems, algebraic geometry, and noncommutative algebra, refers to a pair of finite-dimensional matrices or operators that nearly satisfy the canonical commutation relation. The precise definition depends on the context: in the noncommutative geometry of the Weyl algebra, a CM pair is a pair of matrices (X, Y) such that $[X, Y] + I$ has rank 1; in the general theory of integrable systems, it is the data parametrizing solutions, moduli points, or “spaces” associated with the Calogero–Moser hierarchy and its generalizations. CM pairs play a central organizing role in the paper of the moduli of ideals of the Weyl algebra, the structure of Calogero–Moser spaces, the bispectral problem, and the classification of solutions to certain differential and difference equations.

1. Algebraic and Geometric Definition of Calogero–Moser Pairs

A classical Calogero–Moser pair consists of two $n \times n$ complex matrices $(X, Y)$ satisfying the “rank 1” almost canonical commutation relation:

$[X, Y] + I \quad \text{has rank } 1,$

where $[X, Y] = XY - YX$ and $I$ is the identity matrix. This noncommutative data arises in several independent but equivalent frameworks:

• As moduli points in the Calogero–Moser space
• As objects classifying certain ideals in the Weyl algebra $\mathbb{C}\langle x, y\rangle/(xy - yx - 1)$
• As points parametrizing rank 1 bispectral wave functions for the KP hierarchy

The set of all such pairs modulo the conjugation action of $GL_n(\mathbb{C})$ forms a smooth, irreducible, affine variety of dimension $2n$. The full Calogero–Moser variety is then:

$$\mathcal{C}_n = \left\{ (X, Y) \in M_n(\mathbb{C})^2 : \operatorname{rank}([X, Y] + I) = 1 \right\} / GL_n(\mathbb{C}).$$

The union over $n$ of these spaces plays a central role in noncommutative projective geometry and the bispectral problem (Wilson, 2010).

2. Calogero–Moser Systems and the Moduli of CM Pairs

The Calogero–Moser system, originally a classical many-body integrable system, admits a Lax representation where the Lax matrices are constructed from data determined by CM pairs. The flows of the rational (and trigonometric/elliptic) Calogero–Moser hierarchies preserve the structure of these spaces, and the orbits correspond to isospectral classes of such matrix pairs.

• In the context of complex reflection groups and generalizations, one attaches families of classical and quantum integrable systems (including elliptic Dunkl operator deformations) to group data. The classical CM system is defined on the cotangent bundle $T^*X$ of an abelian variety $X$ (encoded by a finite crystallographic reflection group $G$), and the quantum system is realized as a commuting family of differential operators with poles along reflection hypertori. The commuting integrals (Hamiltonians) are constructed via explicit operator limits of Dunkl operators or as global sections of sheaves of Cherednik algebras (Etingof et al., 2010).

3. Representation-Theoretic and Noncommutative Interpretation

There exists a bijection, uniquely determined by equivariance with respect to the automorphism group $G$ of the Weyl algebra, between:

• Isomorphism classes of noncyclic, rank 1, torsion-free right ideals of the Weyl algebra (the “ideal classes” $R$)
• The union over $n$ of Calogero–Moser varieties $\mathcal{C}_n$ (that is, CM pairs)

This bijection is rigid: any $G$-equivariant map is either the identity (on orbits of fixed $n$) or does not exist between different orbits. Therefore, every geometric or algebraic property preserved by the $G$-action on CM pairs corresponds uniquely to a property of the ideal classes of the Weyl algebra (Wilson, 2010). This rigidity links the structure of the Calogero–Moser varieties directly with noncommutative projective geometry and the classification of right ideals for simple noncommutative algebras.

4. Parametrization and Families of Integrable Systems

In the broader context—especially for elliptic generalizations and systems associated to complex reflection groups—the relevant Calogero–Moser “pairs” are specified by combinatorial or geometric data:

• The family of integrable systems is parametrized by $G$-invariant functions $C$ on pairs $(T, j)$, where $T$ is a reflection hypertorus (the fixed locus of a reflection $s\in G$ in the abelian variety $X$) and $j$ is an integer (often encoding the character of $s$ on the normal bundle). The parameter function $C: \{(T, j)\} \to \mathbb{C}$ determines the residue/coefficient strength of the poles in the Dunkl operators and hence in the CM Hamiltonians.
• For real reflection groups (Weyl groups), this construction recovers classical CM systems; for complex reflection groups, one obtains new integrable systems, often with higher-degree leading symbols in the Hamiltonians (Etingof et al., 2010).

The presence of such parameters encodes generalized CM pairs through geometric and algebraic invariants.

5. Geometric and Representation-Theoretic Cells and Partitions

In the generalization to complex reflection groups and rational Cherednik algebras, one refines the notion of CM pairs by considering:

• Calogero–Moser cells: partitions of the group $W$ into orbits (cells) under inertia groups emerging from the Galois closure and ramification structure of the Cherednik algebra centre. These cells correspond to specific strata in Calogero–Moser spaces (Bonnafé et al., 2012, Bonnafé et al., 2013, Bonnafé et al., 2017).
• Calogero–Moser families and cellular characters: partitions and representation content assigned to irreducible $W$-modules by the action of the Cherednik algebra and its centre. The notion of a “Calogero–Moser pair” can be interpreted as the pair consisting of a point in the parameter space and the associated partition of the set of irreducible representations into such families.
• There is a conjectured equivalence (proved in many cases) between these Calogero–Moser cells/families and the Kazhdan–Lusztig cells and families arising from Hecke algebras for real reflection groups (Bonnafé et al., 2012, Bonnafé et al., 2013, Bonnafé et al., 2017).

6. Applications to Bispectrality, Integrable Models, and Orthogonal Polynomials

CM pairs have further applications in:

• Classification of rank 1 bispectral wave functions for the KP hierarchy, where each such function is associated to a unique Adelio Calogero–Moser pair (Paluso et al., 29 Jul 2025).
• Explicit connection to exceptional Hermite polynomials, where a specific CM pair (X, Z) is constructed so that the generating function for the exceptional family is recovered from a KP wave function built from that pair.
• Construction of characteristic polynomials and spectral data for quantum elliptic Calogero–Moser systems arising in supersymmetric gauge theories, where the role of CM pairs is generalized to polynomials built from partition function data with underlying geometric structure (Chen et al., 2019).

7. Summary Table: Notions of Calogero–Moser Pairs

Notion of Pair	Defining Data	Role in Theory
Matrix Pair (X, Y)	$[X, Y] + I$ has rank 1	Moduli points for $\mathcal{C}_n$; encodes ideals in the Weyl algebra (Wilson, 2010)
Geometric Pair (T, s)	Reflection hypertorus $T$, reflection $s$ used to label poles	Parametrizes integrable CM systems attached to reflection groups (Etingof et al., 2010)
(Parameter, Cell/Family)	Parameter $c$, partition/cell/family in $W$	Organizing data for generalized, Cherednik-type CM models (Bonnafé et al., 2017)
Bispectral Pair	CM pair (X, Z) defining KP wave function	Used in explicit construction of exceptional Hermite polynomials (Paluso et al., 29 Jul 2025)

Each notion provides a different parametrization of data relevant to the definition and application of Calogero–Moser pairs, reflecting the deep interconnections between algebra, geometry, integrable systems, and combinatorics.