Poisson transform and unipotent complex geometry

Our concern is with Riemannian symmetric spaces $Z=G/K$ of the non-compact type and more precisely with the Poisson transform $P_{\lambda }$ which maps generalized functions on the boundary ∂Z to λ-eigenfunctions on Z. Special emphasis is given to a maximal unipotent group $N<G$ which naturally acts on both Z and ∂Z. The N-orbits on Z are parametrized by a torus $A={\left(R_{>0}\right)}^r<G$ (Iwasawa) and letting the level $a\in A$ tend to 0 on a ray we retrieve N via ${\lim }_{a\to 0}Na$ as an open dense orbit in ∂Z (Bruhat). For positive parameters λ the Poisson transform $P_{\lambda }$ is defined and injective for functions $f\in L^2\left(N\right)$ and we give a novel characterization of $P_{\lambda }\left(L^2\right(N\left)\right)$ in terms of complex analysis. For that we view eigenfunctions $\varphi =P_{\lambda }\left(f\right)$ as families ${\left({\varphi }_a\right)}_{a\in A}$ of functions on the N-orbits, i.e. ${\varphi }_a\left(n\right)=\varphi \left(na\right)$ for $n\in N$. The general theory then tells us that there is a tube domain $T=N\exp \left(i\Lambda \right)\subset N_C$ such that each ${\varphi }_a$ extends to a holomorphic function on the scaled tube $T_a=N\exp \left(i\mathrm{Ad}\right(a\left)\Lambda \right)$. We define a class of N-invariant weight functions $w_{\lambda }$ on the tube $T$, rescale them for every $a\in A$ to a weight $w_{\lambda ,a}$ on $T_a$, and show that each ${\varphi }_a$ lies in the $L^2$-weighted Bergman space $B(T_a,w_{\lambda ,a}):=O\left(T_a\right)\cap L^2(T_a,w_{\lambda ,a})$. The main result of the article then describes $P_{\lambda }\left(L^2\right(N\left)\right)$ as those eigenfunctions ϕ for which ${\varphi }_a\in B(T_a,w_{\lambda ,a})$ and$\Vert \varphi \Vert :=\underset{a\in A}{\sup }{a}^{\mathrm{Re}\lambda -2\rho }{\Vert {\varphi }_a\Vert }_{B_{a,\lambda }}<\infty$ holds.