A characterization of the Poisson kernel on the classical real rank one symmetric spaces (Q1210373)

Let \(G\) be a connected real rank one semisimple Lie group and \({\mathfrak g}= {\mathfrak k}+ {\mathfrak a}+ {\mathfrak n}\) be the Iwasawa decomposition for the Lie algebra \({\mathfrak g}\) of \(G\). Let \(K\), \(A\) and \(N\) be the analytic subgroups of \(G\) with Lie algebras \({\mathfrak k}\), \({\mathfrak a}\) and \({\mathfrak n}\) respectively. Then the symmetric space \(G/K\) can be identified with the unit ball \(D^ n= \{x\in \mathbb{F}^ n\); \(\| x\|^ 2 <1\}\), where \(\mathbb{F}= \mathbb{R}\) (resp. \(\mathbb{C}\) and the quaternions \(\mathbb{H}\)) for \(G= \text{SO}_ 0 (n,1)\) (resp. \(\text{SU} (n,1)\) and \(\text{SP} (n,1)\)). For \(x\in G\), we define \(H(x)\) to be the unique element in \({\mathfrak a}\) such that \(x\in K\exp H(x)N\) and define \(\rho(H)= \text{tr} (\text{ad} (H)|_{\mathfrak n})\). Let \(M\) be the centralizer of \(A\) in \(K\) and \(\Delta\) the Laplace-Beltrami operator on \(G/K\). Then the Poisson kernel \(P: G/K\times K/M\to \mathbb{R}\) is defined by \(P(gK, kM)= \exp(-2\rho (H(g^{-1} k)))\). The main results of the present paper are 1) For \(G= \text{SO}_ 0 (n,1)\) and \(\text{SU}(n,1)\), if the real valued \(C^ 2\) function \(F\) on \(G/K\) satisfies (*) \(\Delta F=0\), \(\Delta F^ 2= \lambda F^ 2\), \(F(eK)=1\), then there exists an element \(k_ 0 M\in K/M\), such that \(F(gK)= P(gK, k_ 0 M)\). 2) For \(G= \text{SP} (n,1)\), if the real valued \(C^ 2\) function \(F\) on \(G/K\) satisfies (*) and if (**) there exists an element \(k_ 0 M\in K/M\), such that \(F(k_ 0, w)\) is a function of \(w_ 1+ \overline{w}_ 1\), \(| w_ 1|^ 2\), \(w_ s+ \overline{w}_ s\), \(w_ s i-i \overline{w}_ s\), \(w_ s j-j \overline {w}_ s\), \(w_ s k-k \overline{w}_ s\) (where \(w_ s= x_ s+ ix_{n+s}+ jx_{2n+s}+ kx_{kn+s}\), \(1\leq s\leq n\)), then \(F\) is given by \(F(gK)= P(gK, k_ 0 M)\). The assertion 1) was established elsewhere by author of the present paper and others. However, the method of this paper can be used to treat both the cases in 1) and 2).