Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains (Q1868680)

Let \(U\) and \(W\) be finite-dimensional complex vector spaces, where \(W\) is the complexification of a real vector space \(V\); \(\Omega\) an open convex cone in \(V\); and \(Q: U\times U\rightarrow W\) an \(\Omega\)-positive Hermitian sesquilinear form, i.e., \(Q(u',u)=Q(u,u')^*\) for \(u,u'\in U\) and \(Q(u,u)\in\overline{\Omega}\backslash\{ 0\}\) for all \(u\in U\backslash\{ 0\}\). The Siegel domain \(D=D(\Omega,Q)\) determined by these data is \(D:=\{ (u,w)\in U\times W: w+w^*-Q(u,u)\in\Omega\}\). By \textit{I. I. Piatetskii-Shapiro} [Automorphic functions and the geometry of classical domains. Gordon and Breach, New York (1969; Zbl 0196.09901)], every homogeneous Siegel domain arises in a certain definite way from some normal \(j\)-algebra which is, by definition, a triple \((g, J, \omega)\) of a split solvable Lie algebra \(g\), a linear operator \(J\) on \(g\) with \(J^2=-I\), and a linear form \(\omega\) on \(g\) satisfying the following two conditions: (1) \([Jx,Jy]=[x,y]+J[Jx,y]+J[x,Jy]\) holds for all \(x,y\in g\); (2) \(\langle x|y\rangle_\omega:=\langle [Jx,y],\omega\rangle\) defines a \(J\)-invariant inner product on \(g\). Linear forms \(\omega\) on \(g\) satisfying (2) are said to be admissible. The Koszul form \(\beta\) given by \(\langle x,\beta\rangle :=\operatorname {tr}(\operatorname {ad}(Jx)-J\operatorname {ad}(x))\), \(x\in g\), is an example of admissible linear form by \textit{J. L. Koszul} [Can. J. Math. 7, 562--576 (1955; Zbl 0066.16104), Theorem 1]. The author derives a formula for the action of the Laplace-Beltrami operator \({\mathcal L}_\omega\) corresponding to the metric determined by \(\omega\), on the Poisson kernel for \(D\) transferred to the Lie group \(G=\exp g\). This formula involves an admissible form \(\omega\) and Cayley transforms for \(D\) which were introduced by the author in [Differ. Geom. Appl. 18, 55--78 (2003; Zbl 1023.22007)]. In particular, one has zero in the right-hand side of this formula (i.e., \({\mathcal L}_\omega\) annihilates the Poisson kernel) if and only if a certain equality holds, which is called by the author a geometric norm equality. The author shows that this in turn occurs if and only if the Siegel domain \(D\) is symmetric, and \(\omega |_{[g,g]}\) is a positive number multiple of \(\beta |_{[g,g]}\). The latter is a new version of the theorem of \textit{L. K. Hua} and \textit{K. H. Look} [Sci. Sin. 8, 1031--1094 (1959; Zbl 0090.29503)], \textit{A. Korányi} [Ann. Math. (2) 82, 332--350 (1965; Zbl 0138.06601)] (the ``if'' part), and \textit{Y. Xu} [Sci. Sin., Special Issue II, 80--90 (1979)] (the ``only if'' part), which reveals its geometrical meaning: the harmonicity of the Poisson kernel turns out to be equivalent to the condition that the image of the Shilov boundary \(\Sigma\) of \(D\) under the Cayley transform (see above) should lie on a sphere centered at origin.