Deformations of unitary Howe dual pairs

Ciubotaru, Dan; Bie, Hendrik De; Martino, Marcelo De; Oste, Roy

doi:10.48550/arxiv.2009.05412

Cited by 2 publications

(5 citation statements)

References 28 publications

(49 reference statements)

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“…The definitions of the symmetries and their relations are given for both options in [8] and the different outcomes of double coverings are covered in Corollary A. 4.…”

Section: The Dihedral Dunkl-dirac Equationmentioning

confidence: 99%

“…It is related to the (Pin(N), osp(1|2)) Howe duality. Other deformations of Howe dualities were recently studied in the rational Cherednik algebra context [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…They are often the first non-trivial examples one can hope to consider completely. Recent investigations on the dihedral case include: closed formulas for intertwining operators [7,28], geometric properties of the Calogero-Moser space associated with dihedral groups [2] and the complete descriptions of the deformed unitary Howe dual pairs [4].…”

Section: Introductionmentioning

confidence: 99%

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Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl–Dirac operator

et al. 2022

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“…The definitions of the symmetries and their relations are given for both options in [8] and the different outcomes of double coverings are covered in Corollary A. 4.…”

Section: The Dihedral Dunkl-dirac Equationmentioning

confidence: 99%

“…It is related to the (Pin(N), osp(1|2)) Howe duality. Other deformations of Howe dualities were recently studied in the rational Cherednik algebra context [4,5].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl–Dirac operator

et al. 2022

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“…It is related to the (Pin(N), osp(1|2)) Howe duality. Other deformations of Howe dualities were recently studied in the rational Cherednik algebra context [5,6].…”

Section: Introductionmentioning

confidence: 99%

“…They are often the first non-trivial examples one can hope to consider completely. Recent investigations on the dihedral case include: closed formulas for intertwining operators [8,29], geometric properties of the Calogero-Moser space associated with dihedral groups [2] and the complete descriptions of the deformed unitary Howe dual pairs [5].…”

Section: Introductionmentioning

confidence: 99%

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator

De Bie,

Langlois-Rémillard,

Oste

et al. 2020

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The Dunkl-Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate its symmetry algebra generated by elements supercommuting with the Dunkl-Dirac operator. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl-Dirac equation are given as a realisation of one family of such irreducible unitary representations. CONTENTS 1. Introduction 2. The dihedral Dunkl-Dirac equation 2.1. The Dunkl-Laplace and Dunkl-Dirac operators 2.2. The dihedral Dunkl operators 3. The symmetry algebra of the Dunkl-Dirac operator 3.1. General symmetry algebra for 3D space 3.2. Dihedral 3D symmetry algebra and ladder operators 3.3. Unitary structure 4. Finite-dimensional irreducible representations 4.1. Preliminary general results and idea of the proofs 4.2. Proof of Theorem 4.1 4.3. Proof of Theorem 4.2 5. The monogenic representations 6. Concluding remarks Acknowledgements References Appendix A. Double coverings A.1. Irreducible representations for the odd case A.2. Irreducible representations for the even case

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Deformations of unitary Howe dual pairs

Cited by 2 publications

References 28 publications

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl–Dirac operator

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl–Dirac operator

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator

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