$q$-Analogue of the degree zero part of a rational Cherednik algebra and generalised Van Diejen's Hamiltonians

Technical Summary: q-Analogue of the Degree Zero Part of a Rational Cherednik Algebra and Generalised Van Diejen's Hamiltonians

Problem Statement
Rational Cherednik algebras (RCA) associated with finite Coxeter groups are well-studied objects with deep connections to integrable systems, geometry, and representation theory. Specifically, the degree zero subalgebra of the type $GL_n$ RCA, denoted $H_{\mathfrak{gl}_n}$, is known to be a flat deformation of a crossed product involving the symmetric group and the image of the universal enveloping algebra $U(\mathfrak{gl}_n)$ under the Jordan–Schwinger (oscillator) representation. This algebra is linked to the rational Calogero–Moser system and possesses a non-trivial centre generated by the Euler operator.

The paper addresses the gap in generalising this specific structure to the $q$-deformed setting. While the double affine Hecke algebra (DAHA) of type $GL_n$ serves as the natural $q$-analogue of the full rational Cherednik algebra, the authors seek to define and characterise a specific subalgebra, $H_{\mathfrak{gl}_n}$, within the DAHA that acts as a $q$-analogue of the degree zero part of the RCA. Furthermore, the authors aim to utilise this algebraic structure to construct new integrable systems, specifically generalising the Macdonald–Ruijsenaars system with external Morse potentials (Van Diejen systems) to cases involving two distinct types of interacting particles.

Methodology
The authors employ a constructive algebraic approach within the framework of the double affine Hecke algebra (DAHA) of type $GL_n$, denoted $H_n = H_{n,q,\tau}$, which depends on parameters $q$ and $\tau$.

1. Algebraic Construction:

   • The authors define a subalgebra $H_{\mathfrak{gl}_n} \subset H_n$ generated by the Hecke generators $T_k$, the $Y$-elements $Y_i^{\pm 1}$, and specific operators $e_{ij} = X_i D_j$ (for $i \neq j$).
   • The operators $D_i$ are constructed as $\tau$-deformations of the $q$-Dunkl operators (or $q$-oscillator operators) $d_i$ from the Jordan–Schwinger representation of the quantum group $U_q(\mathfrak{gl}_n)$. Specifically, $D_i$ are defined via conjugation of a central element $D_n$ by products of Hecke generators.
   • The authors establish that $H_{\mathfrak{gl}_n}$ is a flat $\tau$-deformation of the algebra $\mathcal{A} = \mathbb{C}S_n \ltimes \rho(U_q(\mathfrak{gl}_n)/I_q)$, where $\rho$ is the $q$-oscillator representation.

2. Structural Analysis:

   • Defining Relations: The paper derives all defining relations for $H_{\mathfrak{gl}_n}$, including commutation relations between $T_k$ and $e_{ij}$, and relations between $Y_i^{\pm 1}$ and $e_{ij}$. These relations are shown to deform the relations of the algebra $\mathcal{A}$ when $\tau \to 1$.
   • PBW Basis: An explicit Poincaré–Birkhoff–Witt (PBW) basis is constructed for $H_{\mathfrak{gl}_n}$, proving that the algebra is a free module over the ring of Laurent polynomials in $\tau$.
   • Centre and Centralisers: The centre $Z(H_{\mathfrak{gl}_n})$ is identified as being generated by the element $e_Y = \prod_{i=1}^n Y_i$. The authors prove a double centraliser property: $H_{\mathfrak{gl}_n}$ is the centraliser of $e_Y$ in a larger algebra $\mathcal{A}$ (generated by $T_k, X_i, Y_i^{\pm 1}, D_i$), and conversely, the centraliser of $H_{\mathfrak{gl}_n}$ in $\mathcal{A}$ is the centre $Z(H_{\mathfrak{gl}_n})$. This is interpreted as a $q$-analogue of the $(\mathfrak{gl}_n, \mathfrak{gl}_1)$ Howe duality.

3. Integrable Systems Application:

   • The authors introduce a generalised family of commuting elements $D_i^{(l_1, l_2)}$ depending on integer parameters $l_1, l_2$ and complex coefficients $a_j$.
   • By considering symmetric combinations of these operators acting on the space of symmetric Laurent polynomials, they derive families of commuting $q$-difference operators.
   • A "restriction" procedure (taking a limit where the centre of mass is sent to infinity) is applied to these operators to eliminate the dependence on the symmetric group action, yielding explicit Hamiltonians.

Key Contributions and Results

• Definition of $H_{\mathfrak{gl}_n}$: The paper successfully defines a subalgebra $H_{\mathfrak{gl}_n}$ inside the DAHA that serves as the $q$-analogue of the degree zero part of the rational Cherednik algebra. It is shown that in the limit $q \to 1$, this algebra reduces to the known degree zero subalgebra of the RCA.
• Structural Theorems: The authors provide a complete presentation of $H_{\mathfrak{gl}_n}$ with explicit defining relations and a PBW basis. They prove that $H_{\mathfrak{gl}_n}$ is a flat $\tau$-deformation of the crossed product $\mathbb{C}S_n \ltimes \rho(U_q(\mathfrak{gl}_n))$.
• Centre and Duality: The centre is proven to be generated by the single invertible element $e_Y$. The double centraliser property established relates $H_{\mathfrak{gl}_n}$ to the larger algebra $\mathcal{A}$, generalising the classical result where the degree zero part is the centraliser of the Euler operator.
• New Integrable Hamiltonians:
  • The authors derive new commuting $q$-difference operators that generalise the Macdonald–Ruijsenaars system with Morse potentials (Van Diejen systems).
  • Crucially, they extend these systems to include two types of interacting particles. The resulting Hamiltonian (Equation 6.16) describes a system with $N_1$ particles of one type and $N_2$ particles of another, interacting via both intra-type and inter-type potentials, as well as external Morse-type fields.
  • The differential limit ($q \to 1$) of these operators recovers a generalisation of the hyperbolic Calogero–Moser system with Morse potentials to the two-particle-type case, extending previous work by Inozemtsev and Van Diejen.

Significance
The paper claims significance in two primary areas:

1. Algebraic Theory: It fills a structural gap in the theory of Cherednik algebras by rigorously defining and characterising the $q$-analogue of the degree zero subalgebra. This provides a new link between the double affine Hecke algebra, quantum groups, and integrable systems, specifically through the lens of the Jordan–Schwinger representation. The identification of the double centraliser property offers a new perspective on the representation theory of these algebras, analogous to Howe duality.

2. Integrable Systems: The work provides a systematic algebraic method for generating new integrable difference equations. By extending the Van Diejen system to include two distinct types of particles, the authors generalise the deformed Macdonald–Ruijsenaars operators to settings with external fields. This extends the known landscape of integrable models, offering Hamiltonians that possess discrete spectra in the quantum case and preserve integrability through a large family of commuting conserved quantities. The results connect the representation theory of DAHAs directly to the construction of complex many-body quantum systems.