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 Pλ 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=(R>0)r<G (Iwasawa) and letting the level a∈A tend to 0 on a ray we retrieve N via lima→0⁡Na as an open dense orbit in ∂Z (Bruhat). For positive parameters λ the Poisson transform Pλ is defined and injective for functions f∈L2(N) and we give a novel characterization of Pλ(L2(N)) in terms of complex analysis. For that we view eigenfunctions ϕ=Pλ(f) as families (ϕa)a∈A of functions on the N-orbits, i.e. ϕa(n)=ϕ(na) for n∈N. The general theory then tells us that there is a tube domain T=Nexp⁡(iΛ)⊂NC such that each ϕa extends to a holomorphic function on the scaled tube Ta=Nexp⁡(iAd(a)Λ). We define a class of N-invariant weight functions wλ on the tube T, rescale them for every a∈A to a weight wλ,a on Ta, and show that each ϕa lies in the L2-weighted Bergman space B(Ta,wλ,a):=O(Ta)∩L2(Ta,wλ,a). The main result of the article then describes Pλ(L2(N)) as those eigenfunctions ϕ for which ϕa∈B(Ta,wλ,a) and‖ϕ‖:=supa∈A⁡aReλ−2ρ‖ϕa‖Ba,λ<∞ holds.