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 ${\mathcal{P}}_{\hat{I}}»$ which maps generalized functions on the boundary $\partial Z$ to $\hat{I}»$-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 $\hat{I}»$ the Poisson transform ${\mathcal{P}}_{\hat{I}}»$ is defined an injective for functions $f\in L^2(N)$ and we give a novel characterization of ${\mathcal{P}}_{\hat{I}}»(L^2(N))$ in terms of complex analysis. For that we view eigenfunctions $\ddot{I}•={\mathcal{P}}_{\hat{I}}»(f)$ as families $(\ddot{I}{•}_a{)}_{a\in A}$ of functions on the $N$-orbits, i.e. $\ddot{I}{•}_a(n)=\ddot{I}•(na)$ for $n\in N$. The general theory then tells us that there is a tube domain $\mathcal{T}=N\exp (i\hat{I}›)\subset N_{\mathbb{C}}$ such that each $\ddot{I}{•}_a$ extends to a holomorphic function on the scaled tube ${\mathcal{T}}_a=N\exp (i\mathrm{Ad}(a)\hat{I}›)$. We define a class of $N$-invariant weight functions ${\mathbf{w}}_{\hat{I}}»$ on the tube $\mathcal{T}$, rescale them for every $a\in A$ to a weight ${\mathbf{w}}_{\hat{I}»,a}$ on ${\mathcal{T}}_a$, and show that each $\ddot{I}{•}_a$ lies in the $L^2$-weighted Bergman space $\mathcal{B}({\mathcal{T}}_a,{\mathbf{w}}_{\hat{I}»,a}):=\mathcal{O}({\mathcal{T}}_a)\cap L^2({\mathcal{T}}_a,{\mathbf{w}}_{\hat{I}»,a})$. The main result of the article then describes ${\mathcal{P}}_{\hat{I}}»(L^2(N))$ as those eigenfunctions $\ddot{I}•$ for which $\ddot{I}{•}_a\in \mathcal{B}({\mathcal{T}}_a,{\mathbf{w}}_{\hat{I}»,a})$ and $$\Vert \ddot{I}•\Vert :=\underset{a\in A}{\sup }{a}^{\mathrm{Re}\hat{I}»-2\ddot{I}}\Vert \ddot{I}{•}_a{\Vert }_{{\mathcal{B}}_{a,\hat{I}»}}<\infty$$ holds.

27 pages, to appear in Journal of Functional Analysis