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| Format: | Preprint |
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2022
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| Accès en ligne: | https://arxiv.org/abs/2206.14088 |
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| _version_ | 1866912112118333440 |
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| author | Gimperlein, Heiko Krötz, Bernhard Roncal, Luz Thangavelu, Sundaram |
| author_facet | Gimperlein, Heiko Krötz, Bernhard Roncal, Luz Thangavelu, Sundaram |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14088 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Poisson transform and unipotent complex geometry Gimperlein, Heiko Krötz, Bernhard Roncal, Luz Thangavelu, Sundaram Representation Theory 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. |
| title | Poisson transform and unipotent complex geometry |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2206.14088 |