对称空间上Horocyclic Radon变换的么正化
Unitarization of the Horocyclic Radon Transform on Symmetric Spaces
摘要：
本文考虑对偶对$(X,Ξ)$的Radon变换，其中$X=G/K$是非紧对称空间，$Ξ$是$X$的horocycle空间.我们解决的unitarization的问题，被认为是（并解决在某些情况下）由Helgason，即确定一个伪微分算子，使预组成与Radon变换扩展到一个酉算子$\mathcal{Q}\colon L^2(X)\to L_\flat^2(Ξ)$，其中$L_\flat^2(Ξ)$是一个封闭的子空间的$L^2(Ξ)$占Weyl对称性。进一步证明了酉扩张将$G$在$L^2(X)$和$L_\flat^2(Ξ)$上的拟正则表示交织在一起.
英文摘要：
We consider the Radon transform for a dual pair $(X,Ξ)$, where $X=G/K$ is a noncompact symmetric space and $Ξ$ is the space of horocycles of $X$. We address the unitarization problem that was considered (and solved in some cases) by Helgason, namely the determination of a pseudo-differential operator such that the pre-composition with the Radon transform extends to a unitary operator $\mathcal{Q}\colon L^2(X)\to L_\flat^2(Ξ)$, where $L_\flat^2(Ξ)$ is a closed subspace of $L^2(Ξ)$ which accounts for the Weyl symmetries. Furthermore, we show that the unitary extension intertwines the quasi-regular representations of $G$ on $L^2(X)$ and $L_\flat^2(Ξ)$.
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