Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains

In this paper, we present a geometric norm equality involving an admissible linear form ω for the Shilov boundary of a homogeneous Siegel domain D. We prove that the validity of this norm equality is equivalent to the symmetry of D and the reduction of ω essentially to the Koszul form. This, in particular, reveals a geometric reason that the Poisson kernel is annihilated by the Laplace–Beltrami operator if and only if D is symmetric, a theorem due to Hua, Look, Korányi and Xu.