A generalized Gelfand pair attached to 3-step nilpotent Lie group

自由群N_{3,2}上SO（3）不变Schwartz函数的Gelfand变换

Gelfand transforms of SO(3)-invariant Schwartz functions on the free group N_{3,2}

Gelfand对$(K\ltimes N, K)$的谱，其中$N$是幂零群，可以嵌入欧几里得空间。证明了谱上的Schwartz函数一般是$N$上的Schwartz $K$不变函数的Gelfand变换.我们还证明了Gelfand对$(SO(3)\ltimes N_{3,2}, SO(3))$的情况下的逆，其中$N_{3,2}$是自由的两步幂零李群与三个生成元。这扩展了海森堡群的最近结果。

Gelfand 对的特征刻画：拟 Gelfand 对

Characterization of the Gelfand pair: Quasi Gelfand pair

本文研究局部紧群与自同构群紧子群构成的拟 Gelfand 对，在幂零情形已有特征刻画基础上，进一步描述其关联的球面函数。

The Schwartz correspondence for the complex motion group on ${\mathbb C}^2$

If $(G,K)$ is a Gelfand pair, with $G$ a Lie group of polynomial growth and $K$ a compact subgroup of $G$, the Gelfand spectrum $Σ$ of the bi-$K$-invariant algebra $L^1(K\backslash G/K)$ admits natural embeddings into ${\mathbb R}^n$ spaces as a closed subset. For any such embedding, define ${\mathcal S}(Σ)$ as the space of restrictions to $Σ$ of Schwartz functions on ${\mathbb R}^n$. We call Schwartz correspondence for $(G,K)$ the property that the spherical transform is an isomorphism of ${\mathcal S}(K\backslash G/K)$ onto ${\mathcal S}(Σ)$. In all the cases studied so far, Schwartz corresp…

Some harmonic analysis on commutative nilmanifolds

In this work, we consider a family of Gelfand pairs $(K \ltimes N, N)$ (in short $(K,N)$) where $N$ is a two step nilpotent Lie group, and $K$ is the group of orthogonal automorphisms of $N$. This family has a nice analytic property: almost all these 2-step nilpotent Lie group have square integrable representations. In this cases, following Moore-Wolf's theory, we find an explicit expression for the inversion formula of $N$, and as a consequence, we decompose the regular action of $K \ltimes N$ on $L^{2}(N)$. This result completes the analysis carried out by Wolf, where the inversion formula i…

Spherical analysis on homogeneous vector bundles

Given a Lie group $G$, a compact subgroup $K$ and a representation $τ\in\hat K$, we assume that the algebra of $\text{End}(V_τ)$-valued, bi-$τ$-equivariant, integrable functions on $G$ is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the rôle of the algebra of $G$-invariant differential operators on the homogeneous bundle $E_τ$ over $G/K$. In particular, we observe that, under the above assumptions, $(G,K)$ is a Gelfand pair and show that the Gelfand spectrum for the triple $(G,K,τ)$ admits homeomorphic embeddings in $\mathbb C^n$. In…

Beurling-Fourier代数的解析子代数与李群的复化

Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups

本文通过引入解析子代数，将加权Fourier代数的Gelfand谱元素解释为李群复化中特定点上的取值，并分别给出一般连通李群的局部解、连通单连通幂零李群的全局解，以及$ax+b$-群的全局解实例。

李群上的微分算子代数与谱乘子

Algebras of differential operators on Lie groups and spectral multipliers

本文研究了两两可交换的自伴左不变微分算子L_1，.. L_n上的连通李群G.在假设L_1，.生成的代数L_n包含一个加权次强制算子--这是由ter Elst和Robinson（J. Funct.肛门，157（1）：88--163，1998），包括正椭圆算子，次拉普拉斯算子和Rockland算子，我们证明了L_1，…，L_n在L^2（G）上（本质）自伴且强交换.此外，我们执行这样一个系统的运营商的抽象研究，与代数结构和G的表示论，类似于在文献中所做的微分算子代数与Gelfand对。当G具有多项式体积增长时，得到了算子m（L_1，...，L_n）对应于满足某种光滑条件的紧支撑乘子m。对m所要求的光滑阶与G的多项式的多项式的次数增长有关。在G是齐次李群且L_1，. L_n是齐次算子，证明了一个Mihlin-Hörmander型L^p乘子定理，推广了单个Rockland算子的已知结果.进一步，我们考虑了几个齐次群G_j，每个G_j都有自己的算子系，从而发展了一个乘积理论，证明了一个Marcinkiewicz型的乘子定理，不仅关于G_j的直积，而且关于其他的（可能是非齐次的）群，包含G_j的同态像。因此，对于某些多项式增长的非幂零群和某些特殊的次拉普拉斯算子，我们能够改进Alexopoulos（Proc. Amer. Math. Soc.，120（3）：973-979，1994）。