Cartan-Helgason定理、Poisson变换和Furstenberg-Satake紧化

Satake-Furstenberg紧化、矩映射和λ_1

Satake-Furstenberg compactifications, the moment map and λ_1

设G是复半单李群，K是极大紧子群，V是K的不可约表示。用M表示G在P（V）中的唯一闭轨道，用O表示它通过矩映射的图像。对于M上的任意测度，我们构造了一个从G/K的Satake紧化（与V相关联）到K的李代数的映射。对于K-不变测度，该映射是O的凸包络上的Satake紧化的同胚。对于一大类测度，映射的像是凸包络。作为应用，我们得到了Hermite对称空间上任意Kaehler度量下函数的Laplacian第一特征值的精确上界。

Satake-Furstenberg compactifications and gradient map

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let $τ$ be an irreducible representation of $G$ on a complex vector space $V$. Let $h$ be a Hermitian scalar product on $V$ such that $τ(G)$ is compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $μ_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$…

Poisson变换的精确映射性质与Baum-Connes猜想

Sharp mapping properties of Poisson transforms and the Baum-Connes conjecture

本文证明了Poisson变换在分布层上精确的量化类比，展示了其将Sobolev空间有界映射到L^2空间，并证明了与Furstenberg紧化上光滑函数的交换子(compact)的性质，从而验证了Julg关于Baum-Connes猜想的剩余猜想。