AI 中文总结
该研究针对紧李群闭连通子群确定的次拉普拉斯算子，推导热核积分表示、小时间热迹展开等结果，并应用于$\rm SU(N)$块子群得到经典谱。
AI 中文摘要
我们研究由紧李群的闭连通子群确定的次拉普拉斯算子的热核、次黎曼距离与谱泽塔函数。结合Hall反演公式与紧群热核的仿射格展开，我们推导出涉及群指数格的Cartan积分表示。对于两步紧李对，完整的代数小时间热迹展开在指数小余项范围内由两个显式常数$C_{G,L}$和$\beta_{G,L}$确定；该展开决定了所有热系数、约化谱泽塔函数的极点与留数及其在非正整数处的值。对于平流对称子类，我们证明了Carnot-Carathéodory距离的一致垂直渐近性，其首项系数$\frak F_{G,K}(Z)$是有限维奇异值问题的可达最小值。对于单连通两步对，我们得到泽塔正则化行列式的精确分解，分解为局部、格与谱项，并带有指数截断估计。我们将这些结果专门应用于$\boldsymbol{\rm SU}(N)$的块子群，得到经典$\boldsymbol{\rm SU}(2)$与CR球谱。
英文摘要
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 $\mathfrak F_{G,K}(Z)$ 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 $\mathrm{SU}(N)$, recovering the classical $\mathrm{SU}(2)$ and CR sphere spectra.
Comments49 pages, comments are welcome
