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PreprintCC BY 4.0arXiv

The Casimir on the K-fixed tensor square of the minimal representation of G2(2)

  • Hongrui Yuan
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Abstract

We compute the quadratic Casimir on the K-fixed subspace of the Hilbert tensor square of the unitary minimal representation of the split real group G2(2). In an explicit algebraic basis of invariant tensors, the Casimir acts by a three-term recurrence. Normalizing this basis yields a symmetric Jacobi operator whose self-adjoint realization is unitarily equivalent to multiplication by −9/2−8x2 in a Wilson-polynomial model with parameters (3/4,3/4,7/12,11/12). We give the normalized spectral density and the unitary transform. The resulting spectrum is purely absolutely continuous, equals (−∞,−9/2], and has scalar spectral multiplicity one.

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Cite

BibTeXRIS
BibTeX
@misc{yuan2026casimir,
  title = {{The Casimir on the $K$-fixed tensor square of the minimal representation of $G_2(2)$}},
  author = {Hongrui Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24656},
  eprint = {2610.24656},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24656},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Casimir on the K-fixed tensor square of the minimal representation of G_2(2)
AU  - Hongrui Yuan
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24656
UR  - https://arxiv.org/abs/2610.24656
AB  - We compute the quadratic Casimir on the K-fixed subspace of the Hilbert tensor square of the unitary minimal representation of the split real group G_2(2). In an explicit algebraic basis of invariant tensors, the Casimir acts by a three-term recurrence. Normalizing this basis yields a symmetric Jacobi operator whose self-adjoint realization is unitarily equivalent to multiplication by −9/2−8x^2 in a Wilson-polynomial model with parameters (3/4,3/4,7/12,11/12). We give the normalized spectral density and the unitary transform. The resulting spectrum is purely absolutely continuous, equals (−∞,−9/2], and has scalar spectral multiplicity one.
ER  -