The Mathieu Property for Compact Connected Lie Groups
Abstract
Let G be a compact connected Lie group, let ℛ(G) denote its algebra of representative functions, and let ℐG(f)=∫G f(g) d g be normalized Haar integration. We prove that ker ℐG is a Mathieu--Zhao subspace of ℛ(G) if and only if G is a torus. More strongly, for every nonabelian compact connected Lie group G we construct A,P,Q∈ℛ(G), with A≥ 0 and Anot≡ 0, such that all pure moments of P vanish and the marked moments satisfy an exact Pascal-row identity ℐG(QsPm) =cmbinomm−1s−1ℐG(A4m+s)>0 (1≤ s≤ m), where cm=4m(m!)2/(2m+1)!; the same moments vanish for s>m. The construction begins with a homogeneous, phase-balanced polynomial pair on ℂ2. An exact coefficient identity on the Hopf sphere S3 is transferred by orbit averaging to every compactly supported SU(2)-invariant measure on ℂ2. A highest-weight representation and a visible simple-root doublet produce such a measure on every compact simple Lie group. Phase balance gives descent through every central quotient, and pullback from an adjoint simple quotient handles arbitrary nonabelian compact connected groups. The proof is direct and independent of the implication from the Mathieu conjecture to the Jacobian conjecture. The torus direction is the theorem of Duistermaat and van der Kallen.
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BibTeX
@misc{long2026mathieu,
title = {{The Mathieu Property for Compact Connected Lie Groups}},
author = {Christopher D. Long},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16178},
eprint = {2609.16178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16178},
}
RIS
TY - UNPB
TI - The Mathieu Property for Compact Connected Lie Groups
AU - Christopher D. Long
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.16178
UR - https://arxiv.org/abs/2609.16178
AB - Let G be a compact connected Lie group, let ℛ(G) denote its algebra of representative functions, and let ℐ_G(f)=∫_G f(g) d g be normalized Haar integration. We prove that ker ℐ_G is a Mathieu--Zhao subspace of ℛ(G) if and only if G is a torus. More strongly, for every nonabelian compact connected Lie group G we construct A,P,Q∈ℛ(G), with A≥ 0 and Anot≡ 0, such that all pure moments of P vanish and the marked moments satisfy an exact Pascal-row identity ℐ_G(Q^sP^m) =c_mbinomm−1s−1ℐ_G(A^4m+s)>0 (1≤ s≤ m), where c_m=4^m(m!)^2/(2m+1)!; the same moments vanish for s>m. The construction begins with a homogeneous, phase-balanced polynomial pair on ℂ^2. An exact coefficient identity on the Hopf sphere S^3 is transferred by orbit averaging to every compactly supported SU(2)-invariant measure on ℂ^2. A highest-weight representation and a visible simple-root doublet produce such a measure on every compact simple Lie group. Phase balance gives descent through every central quotient, and pullback from an adjoint simple quotient handles arbitrary nonabelian compact connected groups. The proof is direct and independent of the implication from the Mathieu conjecture to the Jacobian conjecture. The torus direction is the theorem of Duistermaat and van der Kallen.
ER -