Embeddings of Spherical Homogeneous Spaces

We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety.     

Finite Homogeneous Metric Spaces

Siberian Mathematical Journal - The authors study the class of finite homogeneous metric spaces and some of its important subclasses that have natural definitions in terms of the metrics and...     

Homogeneous Spaces with Sections

We study homogeneous Riemannian manifolds all of whose geodesics can be mapped by some isometry into a fixed homogeneous, connected, totally geodesic submanifold, called section. We show that these spaces are locally symmetric if the section is two-dimensional and give non-symmetric counterexamples with higher-dimensional sections.Mathematics Subject Classification (2000): 53C35, 53C30, 53C22, 57S15     

齐型空间上的Lipschitz函数空间

给出了齐型空间上Lipschitz函数空间的两个新的等价范数，证明了Lipschitz函数满足与BMO函数类似的Joho-Nirenberg型不等式．     

Coorbits of Quantum Homogeneous Spaces

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.     

齐型空间上的Morrey空间极大算子有界性

本文研究齐型空间上Morrey空间的极大算子的有界性,得到的结果是已知 结果的本质改进和推广.     

Minimal Homogeneous Submanifolds in Euclidean Spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex homogeneous submanifold of C ^N must be totally geodesic.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

Traces on Noncommutative Homogeneous Spaces

We study properties of C*-algebraic deformations of homogeneous spaces G/Γ which are equivariant in the sense that they preserve the natural action of G by left translation. The center is shown to be isomorphic to C(G/G⁰_ρ) for a certain subgroup G⁰_ρ of G, and there is a 1-1 correspondence between normalised traces and probability measures on G/G⁰_ρ. This makes it possible to represent the deformed algebra as operators over L²(G/Γ). Applications to K-theory are also mentioned.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.     

Approximation on Two-point Homogeneous Spaces

This paper is a short research announcement of the results obtained in [BD], which will appear soon.     

Quantum Cohomology of Minuscule Homogeneous Spaces

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov–Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In particular, we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties. DOI: .     

Spherical Maps Of Euclidean Spaces

We classify the maps f: Rp → Rn suchthat ¦fat -fb¦ = ¦fbt -fc¦ whenever ¦a?b¦= ¦b? c¦.     

Weighted Coorbit Spaces and Banach Frames on Homogeneous Spaces

This article is concerned with frame constructions on domains and manifolds. The starting point is a unitary group representation which is square integrable modulo a suitable subgroup and therefore gives rise to a generalized continuous wavelet transform. Then generalized coorbit spaces can be defined by collecting all functions for which this wavelet transform is contained in a weighted L_p-space. Moreover, we show that a judicious discretization of the representation leads to an atomic decomposition and to Banach frames for these coorbit spaces.     

齐型空间上的Morrey空间的算子有界性及其应用

作者建立了一大类次线性算子和交换子在齐型空间上的Morrey空间中的有界性．应用这些结果,作者研究了具有不连续系数的超抛物方程解的局部正则性．     

两点齐性的Finsler流形

该文证明任何一个两点齐性的Finsler流形一定是黎曼流形．证明过程中作者将泛函分析中经典的Mazur定理推广到不一定是绝对齐次的Minkowski空间上．