齐次超空间上的球函数

具有可解稳定子的球面齐性空间上的调和分析

Harmonic analysis on spherical homogeneous spaces with solvable stabilizer

对球面齐性空间G/H，其中G是单连通半单代数群，H是G的连通可解子群，我们计算了G在G/H上齐性线丛的正则截面空间上的表示的谱.

具有极大幂幺子群良商的仿射球面齐性空间

Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup

对于连通半单代数群G的仿射球面齐性空间G/H，考虑G的极大幂幺子群在G/H上的作用的分解态射.我们证明了这个态射是等维的当且仅当G/H的权半群满足一些简单的条件。

球面齐性空间的唯一性

Uniqueness property for spherical homogeneous spaces

设G是连通约化群.回想一下，如果X是正规的，并且G的博雷尔子群在X上有开轨道，则G簇X称为球面簇。对于球面齐次G-空间，我们赋予了某些组合不变量：权格、赋值锥和B-稳定素除子集。证明了具有相同组合不变量的两个球面齐性空间等变同构。进一步，我们展示了如何从这些不变量中恢复G-等变自同构群。

齐性实球面空间的简单紧化与极式分解

Simple compactifications and polar decomposition of homogeneous real spherical spaces

设Z是一个代数齐性空间Z=G/H，它依附于实约化李群G.我们假设Z是实球面，即，极小抛物子群在Z上有开轨。对于这样的空间，我们研究了它们的大尺度几何，并提供了一个极式分解。这是由本文建立的Z的简单紧化的存在性得到的。

非单半单代数群的仿射球面齐性空间的扩张权半群

Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups

连通半单代数群G的齐性空间G/H的扩张权半群刻画了G在G/H上齐性线性丛的正则截面空间（包括G/H上的正则函数空间）上的表示的谱.本文计算了严格不可约仿射球面齐性空间G/H的扩张权半群，其中G是单连通非单半单复代数群，H是它的连通闭子群，在所有情况下，我们还找到了对应于这个半群的不可分解元素的最高权函数.其中，我们的结果完成了半单复代数群的所有严格不可约单连通仿射球面齐性空间的权半群的计算。

齐次向量丛的球面分析

Spherical analysis on homogeneous vector bundles

给定一个李群$G$，一个紧子群$K$和一个表示$τ\in\hat K$，我们假定$G$上的$\text{End}(V_τ)$ -值双$τ$ -等变可积函数的代数是交换的.本文给出了有关球面分析的基本事实，特别强调了$G/K$-不变微分算子代数在$G/K$上的齐次丛PDMATHAMR上的作用.特别是，我们观察到，在上述假设下，$(G,K)$是一个Gelfand对，并表明三重$(G,K,τ)$的Gelfand谱承认同胚嵌入$\mathbb C^n$。在第二部分中，我们更详细地发展了具有$H$幂零的$G=K\ltimes H$的球面分析。特别地，对于$H=\mathbb R^n$和$K\subset SO(n)$以及海森堡群$H_n$和$K\subset U(n)$，我们刻画了给出交换代数的表示$τ\in \hat K$。\end{abstract}

On extended weight monoids of spherical homogeneous spaces

Given a connected reductive complex algebraic group $G$ and a spherical subgroup $H \subset G$, the extended weight monoid $\widehat Λ^+_G(G/H)$ encodes the $G$-module structures on spaces of global sections of all $G$-linearized line bundles on $G/H$. Assuming that $G$ is semisimple and simply connected and $H$ is specified by a regular embedding in a parabolic subgroup $P \subset G$, in this paper we obtain a description of $\widehat Λ^+_G(G/H)$ via the set of simple spherical roots of $G/H$ together with certain combinatorial data explicitly computed from the pair $(P,H)$. As an application…

半齐次树上的正定函数与球面表示

Positive definite functions on semi-homogeneous trees and spherical representations

本文研究了半齐次树上等距群的正定函数与球面表示，通过紧支径向函数的卷积代数确定正定球面函数，并由此构造了一系列酉球面表示，证明了特征值为0的表示在特定条件下平方可积。

Real orbits of complex spherical homogeneous spaces: the split case

We identify the $G(\mathbb R)$-orbits of the real locus $X(\mathbb R)$ of any spherical complex variety $X$ defined over $\mathbb R$ and homogeneous under a split connected reductive group $G$ defined also over $\mathbb R$. This is done by introducing some reflection operators on the set of real Borel orbits of $X(\mathbb R)$. We thus investigate the existence problem for an action of the Weyl group of $G$ on the set of real Borel orbits of $X(\mathbb R)$. In particular, we determine the varieties $X$ for which these operators define an action of the very little Weyl group of $X$ on the set of…

Invariant differential operators on spherical homogeneous spaces with overgroups

We investigate the structure of the ring ${\mathbb D}_G(X)$ of $G$-invariant differential operators on a reductive spherical homogeneous space $X=G/H$ with an overgroup $\widetilde{G}$. We consider three natural subalgebras of ${\mathbb D}_G(X)$ which are polynomial algebras with explicit generators, namely the subalgebra ${\mathbb D}_{\widetilde{G}}(X)$ of $\widetilde{G}$-invariant differential operators on $X$ and two other subalgebras coming from the centers of the enveloping algebras of $\mathfrak g$ and $\mathfrak k$, where $K$ is a maximal proper subgroup of $G$ containing $H$. We show t…

由洛伦兹群截面产生的球面连续小波变换

Spherical Continuous Wavelet Transforms arising from sections of the Lorentz group

为了研究球面连续小波变换（CWT），我们考虑单位球面$S^{n-1},$的共形群，即所谓的真洛伦兹群Spin $^+(1,n),$.我们的方法是基于构造与齐性空间上的平方可积群表示相关的一般相干态的方法。基本的齐性空间是单位球的回转群被适当的回转子群分解的整个群Spin $^+(1,n)$的扩展。这个齐次空间上的截面由子群自旋$(n)$和$ϕ_a(x)=(x-a)(1+ax)^{-1},$型莫比乌斯变换的旋转构成，其中$a$属于单位球的齐次空间上的给定截面。这以一种自然的方式将安托万和范德海斯特的工作推广到各向异性共形膨胀。

Aperiodic order and spherical diffraction, II: Translation bounded measures on homogeneous spaces

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian …

A Paley-Wiener theorem for spherical $p$-adic spaces and Bernstein morphisms

Let $G$ be (the rational points of) a connected reductive group over a local non-archimedean field $F$. In this article we formulate and prove a property of an $F$-spherical homogeneous $G$-space (which in addition satisfies the finite multiplicity property, which is expected to hold for all $F$-spherical homogeneous $G$-spaces) which we call the Paley-Wiener property. This is much more elementary, but also contains much less information, than the recent relevant work of Delorme, Harinck and Sakellaridis (however, it holds for a wider class of spaces). The property results from a parallel cate…

实球面簇的局部结构定理

The local structure theorem for real spherical varieties

设$G$是一个代数实约化群，$Z$是一个实球面$G$ -簇，即它允许极小抛物子群$P$有一个开轨道.证明了$Z$的一个局部结构定理。在最简单的情况下，$Z$是齐次的，该定理提供了一个同构的开放$P$轨道与一个丛$Q \times_L S$。这里$Q$是具有Levi分解$LU$的抛物子群，$S$是$L$的商$D=L/L_n$的齐次空间，其中$L_n$在$L$中正规，使得$D$是紧模中心。

齐次纤维丛在 Schubert 簇中的应用

Applications of Homogeneous Fiber Bundles to the Schubert Varieties

本文利用旗簇上齐次纤维丛框架研究 Schubert 簇与等变嵌入的关系，证明了相关簇的环面性质，识别了奇妙簇，并给出了光滑球面 Schubert 簇的刻画及其局部刚性等应用。

实球面空间的调和分析

Harmonic analysis for real spherical spaces

介绍了实约化李群$G$上的齐性空间$Z=G/H$的基本调和分析和表示论。特别强调的情况下，$Z$是真正的球形。