PreprintCC BY 4.0arXiv
Spherical harmonics and infinitesimal projective transformations
Abstract
Consider the space of C∞-functions on the two-dimensional sphere S2 and its decomposition ⊕ℋm into a direct sum of minimal rotation-invariant spaces. We consider elements of ⊕ℋm as functions of two variables, a nonnegative integer variable m and a complex variable u (a restriction of such function to a set m=k is a polynomial in u of degree ≤ 2k). The group GL (3,ℝ) of all matrices of order 3 acts on S2, so its Lie algebra acts in C∞(S2) by differentiations along vector fields. We construct operators in 𝒲∞ corresponding such differentiations and obtain a collection of differential-difference operators in 𝒲∞. The expressions include shifts m↦ m± 2 and derivatives of order 4 in u.
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BibTeX
@misc{neretin2026spherical,
title = {{Spherical harmonics and infinitesimal projective transformations}},
author = {Yury A. Neretin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28320},
eprint = {2610.28320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28320},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spherical harmonics and infinitesimal projective transformations
AU - Yury A. Neretin
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28320
UR - https://arxiv.org/abs/2610.28320
AB - Consider the space of C^∞-functions on the two-dimensional sphere S^2 and its decomposition ⊕ℋ_m into a direct sum of minimal rotation-invariant spaces. We consider elements of ⊕ℋ_m as functions of two variables, a nonnegative integer variable m and a complex variable u (a restriction of such function to a set m=k is a polynomial in u of degree ≤ 2k). The group GL (3,ℝ) of all matrices of order 3 acts on S^2, so its Lie algebra acts in C^∞(S^2) by differentiations along vector fields. We construct operators in 𝒲^∞ corresponding such differentiations and obtain a collection of differential-difference operators in 𝒲^∞. The expressions include shifts m↦ m± 2 and derivatives of order 4 in u.
ER -