Boundary Kernel Rigidity and Infinite-Dimensional Complex Hyperbolic Representations

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 combin