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From Dirac to Dunkl Operators through Symmetry Reduction

Cristina Sardón

TL;DR

The paper provides a principled derivation of Dirac-Dunkl operators as symmetry reductions of the flat Dirac operator on Euclidean space, by restricting to a fundamental Weyl chamber with the Heckman-Opdam measure and implementing drift and reflection corrections to preserve skew-adjointness. Through Clifford contraction, the Dunkl operators emerge as the unique first-order deformations compatible with reflection symmetry, and their square yields the Dunkl Laplacian, $D_k^2 = -\Delta_k$. Extending to representation liquids, the authors obtain representation-dependent Dirac-Dunkl operators that act on spinor- or matrix-valued functions, recovering bosonic and fermionic Calogero-Moser systems in the scalar and sign representations, and producing multi-component spin-Calogero models for higher-dimensional representations. The framework unifies analytic, geometric, and representation-theoretic aspects of Dirac and Dunkl operators under a single symmetry-reduction principle, with explicit constructions for types $A_2$ and $A_n$ illustrating Calogero-type realizations across representations.

Abstract

This paper presents a geometric and analytic derivation of Dirac-Dunkl operators as symmetry reductions of the flat Dirac operator on Euclidean space. Starting from the standard Dirac operator, we restrict to a fundamental Weyl chamber of a finite Coxeter group equipped with the Heckman-Opdam measure, and determine the necessary drift and reflection corrections that ensure formal skew-adjointness under this weighted geometry. This procedure naturally reproduces the Dunkl operators as the unique first-order deformations compatible with reflection symmetry, whose Clifford contraction defines the Dirac-Dunkl operator and whose square yields the Dunkl Laplacian. We then extend the construction to include arbitrary unitary representations of the reflection group, obtaining representation-dependent Dirac-Dunkl operators that act on spinor- or matrix-valued functions. In the scalar and sign representations, these operators recover respectively the bosonic and fermionic Calogero-Moser systems, while higher-dimensional representations give rise to multi-component spin-Calogero models. The resulting framework unifies analytic, geometric, and representation-theoretic aspects of Dirac and Dunkl operators under a single symmetry-reduction principle.

From Dirac to Dunkl Operators through Symmetry Reduction

TL;DR

The paper provides a principled derivation of Dirac-Dunkl operators as symmetry reductions of the flat Dirac operator on Euclidean space, by restricting to a fundamental Weyl chamber with the Heckman-Opdam measure and implementing drift and reflection corrections to preserve skew-adjointness. Through Clifford contraction, the Dunkl operators emerge as the unique first-order deformations compatible with reflection symmetry, and their square yields the Dunkl Laplacian, . Extending to representation liquids, the authors obtain representation-dependent Dirac-Dunkl operators that act on spinor- or matrix-valued functions, recovering bosonic and fermionic Calogero-Moser systems in the scalar and sign representations, and producing multi-component spin-Calogero models for higher-dimensional representations. The framework unifies analytic, geometric, and representation-theoretic aspects of Dirac and Dunkl operators under a single symmetry-reduction principle, with explicit constructions for types and illustrating Calogero-type realizations across representations.

Abstract

This paper presents a geometric and analytic derivation of Dirac-Dunkl operators as symmetry reductions of the flat Dirac operator on Euclidean space. Starting from the standard Dirac operator, we restrict to a fundamental Weyl chamber of a finite Coxeter group equipped with the Heckman-Opdam measure, and determine the necessary drift and reflection corrections that ensure formal skew-adjointness under this weighted geometry. This procedure naturally reproduces the Dunkl operators as the unique first-order deformations compatible with reflection symmetry, whose Clifford contraction defines the Dirac-Dunkl operator and whose square yields the Dunkl Laplacian. We then extend the construction to include arbitrary unitary representations of the reflection group, obtaining representation-dependent Dirac-Dunkl operators that act on spinor- or matrix-valued functions. In the scalar and sign representations, these operators recover respectively the bosonic and fermionic Calogero-Moser systems, while higher-dimensional representations give rise to multi-component spin-Calogero models. The resulting framework unifies analytic, geometric, and representation-theoretic aspects of Dirac and Dunkl operators under a single symmetry-reduction principle.

Paper Structure

This paper contains 13 sections, 6 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Fundamentals
  3. Dirac-Dunkl operators and the flat Dirac operator
  4. Reduction of flat Dirac operators by finite Coxeter groups
  5. Drift compensation
  6. Reflection correction
  7. Dirac--Dunkl construction
  8. Global $W$--equivariant formulation
  9. Dirac-Dunkl Operators with Nontrivial Representations
  10. Explicit Calogero--type realizations for type $A_2$.
  11. Global formulation of $A_n$ Dirac-Dunkl operators.
  12. Explicit Calogero--type realization for type $A_n$.
  13. Conclusions

Key Result

Proposition 1

For $\xi \in V$, the adjoint of the directional derivative $\partial_\xi$ with respect to the weighted inner product $\langle \cdot , \cdot \rangle_k$ is given by ignoring boundary terms. In particular,

Theorems & Definitions (16)

  • Remark 1: Basis independence
  • Remark 2
  • Proposition 1
  • proof
  • Remark 3
  • Proposition 2: Drift compensator
  • proof
  • Proposition 3: Reflection compensator
  • proof
  • Definition 1: Dirac--Dunkl operator
  • ...and 6 more