PreprintCC BY 4.0arXiv
Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations
Abstract
Consider the space of C∞-functions on the two-dimensional sphere S2 and its decomposition ⊕ℋn into a direct sum of minimal rotation-invariant spaces. We consider elements of ⊕ℋn as functions of two variables, a nonnegative integer variable n and a complex variable u (a restriction of such function to the set n=k is a polynomial in u of degree ≤ 2k). For operators of multiplication by x1, x2, x3 in C∞(S2) we obtain the corresponding operators in ⊕ℋn, they are differential-difference operators in the variables u, n (including second derivatives in u and shifts n↦ n±1). We obtain the similar correspondence for operators of differentiation along conformal vector fields on S2.
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BibTeX
@misc{neretin2026spherical,
title = {{Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations}},
author = {Yury A. Neretin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27570},
eprint = {2609.27570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27570},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations
AU - Yury A. Neretin
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27570
UR - https://arxiv.org/abs/2609.27570
AB - Consider the space of C^∞-functions on the two-dimensional sphere S^2 and its decomposition ⊕ℋ_n into a direct sum of minimal rotation-invariant spaces. We consider elements of ⊕ℋ_n as functions of two variables, a nonnegative integer variable n and a complex variable u (a restriction of such function to the set n=k is a polynomial in u of degree ≤ 2k). For operators of multiplication by x_1, x_2, x_3 in C^∞(S^2) we obtain the corresponding operators in ⊕ℋ_n, they are differential-difference operators in the variables u, n (including second derivatives in u and shifts n↦ n±1). We obtain the similar correspondence for operators of differentiation along conformal vector fields on S^2.
ER -