Poisson transform and unipotent complex geometry

Abstract: Our concern is with Riemannian symmetric spaces $Z=G/K$ of the non-compact type and more precisely with the Poisson transform $\mathcal{P}\lambda$ which maps generalized functions on the boundary $\partial Z$ to $\lambda$-eigenfunctions on $Z$. Special emphasis is given to a maximal unipotent group $N<G$ which naturally acts on both $Z$ and $\partial Z$. The $N$-orbits on $Z$ are parametrized by a torus $A=(\mathbb{R}{>0})^r<G$ (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 $\partial Z$ (Bruhat). For positive parameters $\lambda$ the Poisson transform $\mathcal{P}\lambda$ is defined an injective for functions $f\in L^2(N)$ and we give a novel characterization of $\mathcal{P}\lambda(L^2(N))$ in terms of complex analysis. For that we view eigenfunctions $\phi = \mathcal{P}\lambda(f)$ as families $(\phi_a){a\in A}$ of functions on the $N$-orbits, i.e. $\phi_a(n)= \phi(na)$ for $n\in N$. The general theory then tells us that there is a tube domain $\mathcal{T}=N\exp(i\Lambda)\subset N_\mathbb{C}$ such that each $\phi_a$ extends to a holomorphic function on the scaled tube $\mathcal{T}a=N\exp(i\operatorname{Ad}(a)\Lambda)$. We define a class of $N$-invariant weight functions ${\bf w}\lambda$ on the tube $\mathcal{T}$, rescale them for every $a\in A$ to a weight ${\bf w}{\lambda, a}$ on $\mathcal{T}_a$, and show that each $\phi_a$ lies in the $L^2$-weighted Bergman space $\mathcal{B}(\mathcal{T}_a, {\bf w}{\lambda, a}):=\mathcal{O}(\mathcal{T}a)\cap L^2(\mathcal{T}_a, {\bf w}{\lambda, a})$. The main result of the article then describes $\mathcal{P}\lambda(L^2(N))$ as those eigenfunctions $\phi$ for which $\phi_a\in \mathcal{B}(\mathcal{T}_a, {\bf w}{\lambda, a})$ and $$|\phi|:=\sup_{a\in A} a^{\operatorname{Re}\lambda -2\rho} |\phi_a|{\mathcal{B}{a,\lambda}}<\infty$$ holds.