Boundary Kernel Rigidity and Infinite-Dimensional Complex Hyperbolic Representations

Abstract

For n>=2, we prove a rigidity theorem for the Hermitian boundary kernels L_{t,s}(x,y)=|1- |^t exp(is arg(1- )), 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.