Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants
Abstract
We study heat kernels, sub-Riemannian distances, and spectral zeta functions of sub-Laplacians determined by closed connected subgroups of compact Lie groups. Combining Hall's inversion formula with the affine lattice expansion of the compact group heat kernel, we derive a Cartan integral representation involving the group's exponential lattice. For two-step compact Lie pairs, the full algebraic small-time heat trace expansion is determined, up to an exponentially small remainder, by two explicit constants CG,L and βG,L. This expansion determines all heat coefficients, the poles and residues of the reduced spectral zeta function, and its values at nonpositive integers. For the transvective symmetric subclass, we prove uniform vertical asymptotics for the Carnot-Carathéodory distance; the leading coefficient FG,K(Z) is the attained minimum of a finite-dimensional singular value problem. For simply connected two-step pairs, we obtain an exact decomposition of the zeta-regularized determinant into local, lattice, and spectral terms, with exponential truncation estimates. We specialize these results to block subgroups of SU(N), recovering the classical SU(2) and CR sphere spectra.
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BibTeX
@misc{bauer2026sublaplacians,
title = {{Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants}},
author = {Wolfram Bauer and Zhicheng Han and Zhipeng Yang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31070},
eprint = {2608.31070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31070},
}
RIS
TY - UNPB
TI - Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants
AU - Wolfram Bauer
AU - Zhicheng Han
AU - Zhipeng Yang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31070
UR - https://arxiv.org/abs/2608.31070
AB - We study heat kernels, sub-Riemannian distances, and spectral zeta functions of sub-Laplacians determined by closed connected subgroups of compact Lie groups. Combining Hall's inversion formula with the affine lattice expansion of the compact group heat kernel, we derive a Cartan integral representation involving the group's exponential lattice. For two-step compact Lie pairs, the full algebraic small-time heat trace expansion is determined, up to an exponentially small remainder, by two explicit constants C_G,L and β_G,L. This expansion determines all heat coefficients, the poles and residues of the reduced spectral zeta function, and its values at nonpositive integers. For the transvective symmetric subclass, we prove uniform vertical asymptotics for the Carnot-Carathéodory distance; the leading coefficient F_G,K(Z) is the attained minimum of a finite-dimensional singular value problem. For simply connected two-step pairs, we obtain an exact decomposition of the zeta-regularized determinant into local, lattice, and spectral terms, with exponential truncation estimates. We specialize these results to block subgroups of SU(N), recovering the classical SU(2) and CR sphere spectra.
ER -