论文标题
泊松变换和一体式复合几何形状
Poisson transform and unipotent complex geometry
论文作者
论文摘要
我们关注的是，riemannian对称空间$ z = g/k $的非连接类型，更准确地说是Poisson Transform $ \ MATHCAL {p}_λ$，在$ z $上映射了边界$ \ partial z $至$λ$ eigenfunctions上的边界$ \ partial z $上的广义函数。特别强调了最大的一体组$ n <g $，该$ n <g $自然地对$ z $和$ \ partial z $。 $ z $上的$ n $ -orbits由torus $ a =（\ mathbb {r} _ {> 0}）^r <g $（iwasawa），让$ a \ in $ a \ in a $ ty $ 0 $ to $ 0 $在ray上我们通过$ n $ na $ na \ z a n $ na \ n na $ n na $ n n $ na \ t na $ na \ t na $ na \ e the （Bruhat）。对于正参数，$λ$ poisson变换$ \ mathcal {p}_λ$被定义为功能的注入$ f \ in l^2（n）$中，我们在复杂分析中给出了$ \ mathcal {p}_λ（l^2（n））$的新颖表征。为此，我们将eigenfunctions $ ϕ = \ mathcal {p}_λ（f）$作为famess $（ϕ_a）_ {a \ in a} $ n $ n $ -orbits上的a} $，即$ n $ -orbits，即$ n $ na（n）= ϕ（na）= ϕ（na）$ nna = na）$ n $ n $ n $ n $ n $。然后，一般理论告诉我们，有一个管域$ \ MATHCAL {t} = n \ exp（iλ）\ subset n_ \ mathbb {c} $，这样每个$ ϕ_A $都会扩展到缩放的管$ \ mathcal {t} _a _a = n \ exp（a f perticeard）上的全体形状功能。 We define a class of $N$-invariant weight functions ${\bf w}_λ$ on the tube $\mathcal{T}$, rescale them for every $a\in A$ to a weight ${\bf w}_{λ, a}$ on $\mathcal{T}_a$, and show that each $ϕ_a$ lies in the $ l^2 $ - 加权伯格曼空间$ \ mathcal {b}（\ Mathcal {t} _a，{\ bf w} _ {λ，a}）：= \ nathcal {o {o}（o {o {o） w} _ {λ，a}）$。然后，本文的主要结果将$ \ MATHCAL {p}_λ（l^2（n））$作为那些eigenfunctions $ ϕ $ in \ in \ in \ nathcal {b}（\ Mathcal {\ Mathcal {t} _a $ \ | ϕ \ |：= \ sup_ {a \ in} a} a^{\ operatatorName {re}λ-2ρ} \ | ϕ_a \ | _ {\ Mathcal {b} _ {b} _ {a {a {a，λ}}}} <\ iffty $$持有。
Our concern is with Riemannian symmetric spaces $Z=G/K$ of the non-compact type and more precisely with the Poisson transform $\mathcal{P}_λ$ which maps generalized functions on the boundary $\partial Z$ to $λ$-eigenfunctions on $Z$. Special emphasis is given to a maximal unipotent group $N<G$ which naturally acts on both $Z$ and $\partial Z$. The $N$-orbits on $Z$ are parametrized by a torus $A=(\mathbb{R}_{>0})^r<G$ (Iwasawa) and letting the level $a\in A$ tend to $0$ on a ray we retrieve $N$ via $\lim_{a\to 0} Na$ as an open dense orbit in $\partial Z$ (Bruhat). For positive parameters $λ$ the Poisson transform $\mathcal{P}_λ$ is defined an injective for functions $f\in L^2(N)$ and we give a novel characterization of $\mathcal{P}_λ(L^2(N))$ in terms of complex analysis. For that we view eigenfunctions $ϕ= \mathcal{P}_λ(f)$ as families $(ϕ_a)_{a\in A}$ of functions on the $N$-orbits, i.e. $ϕ_a(n)= ϕ(na)$ for $n\in N$. The general theory then tells us that there is a tube domain $\mathcal{T}=N\exp(iΛ)\subset N_\mathbb{C}$ such that each $ϕ_a$ extends to a holomorphic function on the scaled tube $\mathcal{T}_a=N\exp(i\operatorname{Ad}(a)Λ)$. We define a class of $N$-invariant weight functions ${\bf w}_λ$ on the tube $\mathcal{T}$, rescale them for every $a\in A$ to a weight ${\bf w}_{λ, a}$ on $\mathcal{T}_a$, and show that each $ϕ_a$ lies in the $L^2$-weighted Bergman space $\mathcal{B}(\mathcal{T}_a, {\bf w}_{λ, a}):=\mathcal{O}(\mathcal{T}_a)\cap L^2(\mathcal{T}_a, {\bf w}_{λ, a})$. The main result of the article then describes $\mathcal{P}_λ(L^2(N))$ as those eigenfunctions $ϕ$ for which $ϕ_a\in \mathcal{B}(\mathcal{T}_a, {\bf w}_{λ, a})$ and $$\|ϕ\|:=\sup_{a\in A} a^{\operatorname{Re}λ-2ρ} \|ϕ_a\|_{\mathcal{B}_{a,λ}}<\infty$$ holds.
