AI 中文总结
该论文证明了边界核刚性定理，并据此分类了无限维复双曲空间中 PU(n,1) 的不可约表示及全等距群的自表示，揭示了平移因子与 Cartan 因子的符号关系。
AI 中文摘要
对于 n>=2，我们证明了 Hermitian 边界核 L_{t,s}(x,y)=|1-<x,y>|^t exp(is arg(1-<x,y>))（x != y）在 S^{2n-1} 上的刚性定理，其中 L_{t,s}(x,x)=0，t>0，且 s 属于 [-1,1]。L_{t,s} 的每个有限 Gram 矩阵的正指数至多为 1，当且仅当 0<t<=1 且 s=+-t。证明结合了边界圆上的傅里叶分析与对两个正交复方向的限制。维度阈值是精确的。通过对等变边界映射的 Gram 核进行归一化，我们证明了从 PU(n,1) 到无限维复双曲空间的全纯等距群的每个连续不可约表示的平移长度因子 t 和带符号的 Cartan 因子 s 满足 s=+-t。将此与已知的端点刚性、Monod 的构造以及 Ruiz Stolowicz 的完全不变量定理相结合，得到分类：全纯共轭类由 (0,1) x {+-1} 中的参数 (t,epsilon) 参数化。允许反全纯共轭将两个符号等同。利用 Monod 的自动连续性论证和相同的边界刚性，我们还分类了无限维复双曲空间的完全等距群的所有不可约自表示。在共轭意义下，这些恰好是参数为 0<t<=1 的 Monod 表示。特别地，Monod 分类定理中的典范轨道假设是自动成立的。
英文摘要
For n>=2, we prove a rigidity theorem for the Hermitian boundary kernels L_{t,s}(x,y)=|1-<x,y>|^t exp(is arg(1-<x,y>)), x != y, on S^{2n-1}, with L_{t,s}(x,x)=0, t>0, and s in [-1,1]. Every finite Gram matrix of L_{t,s} has positive index at most one if and only if 0<t<=1 and s=+-t. The proof combines Fourier analysis on boundary circles with restrictions to two orthogonal complex directions. The dimension threshold is sharp. By normalizing the Gram kernels of equivariant boundary maps, we show that the translation-length factor t and the signed Cartan factor s of every continuous irreducible representation from PU(n,1) to the holomorphic isometry group of infinite-dimensional complex hyperbolic space satisfy s=+-t. Combining this with the known endpoint rigidity, Monod's constructions, and Ruiz Stolowicz's complete-invariant theorem yields the classification: the holomorphic conjugacy classes are parametrized by (t,epsilon) in (0,1) x {+-1}. Allowing anti-holomorphic conjugacy identifies the two signs. Using Monod's automatic-continuity argument and the same boundary rigidity, we also classify all irreducible self-representations of the full isometry group of infinite-dimensional complex hyperbolic space. Up to conjugacy, these are precisely Monod's representations with parameter 0<t<=1. In particular, the canonical-orbit hypothesis in Monod's classification theorem is automatic.
CommentsThe proof of Theorem 1.1 in Section 3 may contain an error. We cannot presently verify the claimed boundary-kernel rigidity, on which the classification results in Theorems 1.2 and 1.3 depend. We therefore withdraw this version and advise readers not to rely on its main conclusions
