Trigonometric Heckman-Opdam Polynomials

• Trigonometric Heckman-Opdam polynomials are defined via Gram-Schmidt orthogonalization of orbit-sum bases and serve as joint eigenfunctions of commuting trigonometric Cherednik operators.
• They generalize classical Jacobi orthogonal polynomials to symmetric spaces, providing key tools in harmonic analysis, integrable systems, and stochastic processes.
• Their vector-valued extensions, arising from parabolic subgroup invariance, enable structured recurrence relations and deep connections with representation theory.

Trigonometric Heckman-Opdam polynomials comprise a fundamental class of multivariable special functions associated with reduced root systems, equipped with $W$-invariant multiplicity parameters. They generalize classical (Jacobi) orthogonal polynomials to symmetric spaces and their degenerations, appearing as simultaneous eigenfunctions of commutative algebras of trigonometric Cherednik operators. Both scalar-valued and vector-valued extensions exist, with the latter arising naturally via invariance under parabolic subgroups of finite reflection groups. Trigonometric Heckman-Opdam polynomials are intrinsically linked to harmonic analysis, representation theory, stochastic processes, and integrable systems.

1. Algebraic and Root-Theoretic Framework

Let $\mathfrak{a}$ be a finite-dimensional real Euclidean space with inner product $(\cdot,\cdot)$, and $R\subset \mathfrak{a}^*$ a reduced root system with Weyl group $W$. The multiplicity function $k_\alpha\ge0$ is required to be $W$-invariant. The weight lattice $P\subset\mathfrak{a}^*$ and the coroot lattice $Q^\vee$ give rise to the compact torus

$$A = \mathfrak{a}/(2\pi i Q^\vee) \simeq \{\,t=e^\lambda : \lambda\in P\,\}$$

on whose algebra of Laurent polynomials $\mathbb{C}[P]$ the analysis is carried out. For a subset $I\subset S$ ($S$ the set of simple reflections), the parabolic subgroup $W_I$ is generated by the reflections associated to $I$. The corresponding $W_I$-invariant Laurent polynomials $\mathbb{C}[P]^{W_I}$ form the natural module for vector-valued trigonometric Heckman-Opdam polynomials (Pruijssen, 2023).

2. Trigonometric Cherednik Operators and Commuting Algebra

Given $\xi\in\mathfrak{h}=\mathfrak{a}\otimes\mathbb{C}$, the (trigonometric) Cherednik operator acts on $\mathbb{C}[P]$ as

$$D_\xi(k) = \partial_\xi + \sum_{\alpha>0} k_\alpha\,\alpha(\xi)\,\frac{1}{1-e^{-\alpha}}(1-s_\alpha) - \rho(k)(\xi),$$

where $\partial_\xi$ is the directional derivative and $\rho(k) = \frac12\sum_{\alpha>0} k_\alpha\alpha$. The Cherednik operators commute: $[D_\xi(k), D_\eta(k)] = 0,$ and are symmetric with respect to the canonical Hermitian inner product

$$(\phi,\psi)_k = \int_T \overline{\phi(t)}\,\psi(t)\,\delta_k(t)\,dt, \qquad \delta_k(t) = \prod_{\alpha\in R^+} (e^{\alpha/2} - e^{-\alpha/2})^{2k_\alpha}.$$

Functional calculus yields the commutative algebra $$S'(\mathfrak{h}) = \mathbb{C}[D_\xi(k):\xi\in\mathfrak{h}]$$, and its $W_I$-invariants $S'(\mathfrak{h})^{W_I}$ result in a family of commuting differential-reflection operators $D_{I,q}$ acting on $\mathbb{C}[P]^{W_I}$ (Pruijssen, 2023).

3. Definition and Spectral Properties

The vector-valued (parabolic) trigonometric Heckman-Opdam polynomials $p_I(\lambda, k)$ are defined via Gram-Schmidt orthogonalization on the Steinberg (orbit-sum) basis

$$m_I(\lambda) = \sum_{\mu\in W_I\cdot \lambda} e^\mu, \qquad \lambda\in P_I^+,$$

with respect to a partial order $\leq_I$ on $P_I^+$. The resulting unique up-to-scaling family satisfies

$$p_I(\lambda, k) = m_I(\lambda) + \sum_{\mu<_I\lambda}c_{\lambda,\mu}m_I(\mu); \qquad (p_I(\lambda, k), m_I(\mu))_k=0\ \forall \mu<_I\lambda.$$

Equivalent characterizations involve non-symmetric Heckman-Opdam polynomials $E(\lambda,k)$: $$p_I(\lambda,k) = \sum_{w\in W_I^\lambda} w\,E(\lambda,k).$$ Each $p_I(\lambda,k)$ is a simultaneous eigenfunction of the commuting algebra $D_{I,q}$: $$D_{I,q}\,p_I(\lambda,k) = q(\widetilde{\lambda})p_I(\lambda,k),\qquad\widetilde{\lambda} = \lambda - v(\lambda)^{-1}\rho(k)\in\mathfrak{h}^*,$$ with orthogonality following from the injectivity and spectral separation properties of these operators (Pruijssen, 2023).

4. Scalar, Non-symmetric, and Vector-Valued Polynomials

Specializing to $I=\emptyset$ gives non-symmetric Heckman-Opdam polynomials. For $I=S$ (the full Weyl group), one recovers the classical $W$-symmetric (scalar-valued) Jacobi polynomials of Heckman and Opdam, which are the $W$-invariant joint eigenfunctions. The full range of $I$ thus interpolates between non-symmetric, vector-valued, and symmetric cases.

For type $A_1$ ($\mathbb{C}[P]\cong \mathbb{C}[e^{\pm x}]$), the symmetric polynomials $p_S(nx, k)$ reduce, up to normalization, to classical Jacobi polynomials (Gegenbauer) or Gauss hypergeometric functions. In higher rank, explicit vector-valued families can be constructed; for instance, type $A_2$ with middle parabolic $I=\{s_2\}$ leads to $3\times3$ matrix-valued differential operators coinciding after conjugation with radial operators of compact symmetric spaces of type $A_2$ (Pruijssen, 2023).

5. Recurrence, Triangularity, and Orthogonality

Triangularity in the Steinberg or monomial basis uniquely specifies the family $p_I(\lambda,k)$. Their orthogonality is a consequence of the spectral properties of the algebra $S'(\mathfrak{h})^{W_I}$. Multiplication by $W_I$-invariant Laurent polynomials yields structured, banded recurrence relations. For the rank-one case, this reduces to three-term Jacobi recurrences; in higher rank, complex multi-index recurrences control raising/lowering in root directions. In all cases, normalization of $p_I(\lambda,k)$ can be set so that the leading coefficient is $1$, or so that the $L^2$ norm is unity, with alternate conventions used in spherical-function or representation-theoretic contexts (Pruijssen, 2023).

6. Explicit Examples and Connections to Other Polynomial Families

In the classical rank-one case ($A_1$), trigonometric Heckman-Opdam polynomials are explicitly linked to Jacobi and ultraspherical (Gegenbauer) polynomials, with associated Poisson kernels, product formulas, convolved translation structures, and fractional integrals (Amri et al., 14 Dec 2025). For $C_2$, four families of two-variable orthogonal polynomials arise, of which the symmetric $P^+_{(\lambda,\mu)}$ family recovers the trigonometric Heckman-Opdam polynomials for root multiplicity $(1,1)$. For general multiplicity, deformation of the Weyl-orbit sum by continuous powers yields the full trigonometric Heckman-Opdam polynomials (Motlochova et al., 2011).

Integral representations via Harish-Chandra-type and Gelfand-Tsetlin polytope integrals are available in all ranks (Sun, 2014). The orthogonality is governed by appropriate torus-invariant measures, with closed form norm formulas (Macdonald-Mehta) and kernel expansions (Cauchy- or determinantal-type identities) (Rösler et al., 2021).

7. Applications and Extensions

Trigonometric Heckman-Opdam polynomials play a central role in harmonic analysis on symmetric spaces, representation theory of real and $p$-adic groups, random matrix theory, and integrable systems. They interpolate between spherical functions, Jack and Macdonald polynomials, and form the analytic backbone of the $\beta$-ensembles and Calogero-Moser-Sutherland theories. In stochastic analysis, they yield eigenfunctions of diffusion generators (Dunkl-Cherednik operators), enabling the explicit construction of martingales and functional expectations, and in combinatorics and spectral theory through their determinantal and recursive structures (Rösler et al., 2021). The extension to vector-valued and matrix-valued polynomials provides a robust framework for studying higher $K$-types and multivariable special function theory, with generalization prospects across harmonic analysis and algebra (Pruijssen, 2023).