Eigenspaces of the Laplacian on hyperbolic spaces: Composition series and integral transforms
Eigenspaces of the Laplacian on hyperbolic spaces: Composition series and integral transforms
复制标题
双曲空间上拉普拉斯算子的特征空间：复合级数和积分变换
DOI：
10.1016/0022-1236(87)90130-3
复制
发表时间：
1987
影响因子：
1.7
通讯作者：
H. Schlichtkrull
中科院分区：
文献类型：
--
作者：
H. Schlichtkrull
设X是射影实数、复数或四元数双曲空间，实现为伪黎曼对称空间X≅G H，其中G=O(p，q)，U(p，q)或Sp(p，q)(这些都是经典的迷向对称空间)。设Δ是X上的G-不变Laplace-Beltrami算子，对每个χ∈C，给出了特征空间{f∈C∞(X)？Δf=χf}的所有闭G-不变子空间的完整刻画.利用“泊松变换”将特征空间表示法与主级数表示法进行了比较。对于例外的各向同性对称空间，也得到了类似的结果。确定了球面离散级数表示的朗兰兹参数。
Let X be a projective real, complex, or quaternion hyperbolic space, realized as the pseudo-Riemannian symmetric space X≅ G H with G= O (p, q), U (p, q), or Sp (p, q)(these are the classical isotropic symmetric spaces). Let Δ be the G-invariant Laplace-Beltrami operator on X. A complete description (by K-types), for each χ∈ C, of all closed G-invariant subspaces of the eigenspace {f∈ C∞(X)¦ Δf= χf} is given. The eigenspace representations are compared with principal series representations, using “Poisson-transformations”. Similar results are obtained also for the exceptional isotropic symmetric space. The Langlands parameters of the spherical discrete series representations are determined.