指数李群上的不变微分算子和中心Fourier乘子
Invariant differential operators and central Fourier multipliers on exponential Lie groups
摘要：
设$G$是指数可解李群。根据定义，$G$是$\ast$-正则的，如果对于$G$的所有酉表示$π$，$ker_{L^1(G)}π$在$ker_{C^\ast(G)}π$中是稠密的。Boidol用纯代数条件刻画了$\ast$-正则指数李群.在本文中，我们将重点关注非$\ast$-正则群。我们说$G$是本原$\ast$-正则的，如果对所有的不可约表示都满足上述密度条件.我们的目标是开发适当的工具来验证这种较弱的属性。为此，我们将介绍Duflo对$(W,p)$和中央傅立叶乘法器$ψ$上的稳定器$M=G_fN$的代表$π=\mcK(f)$在一般的位置。利用Littlewood-Paley理论，我们将得到一些关于乘子算子$T_ψ$的结果，这些结果可能是独立有趣的。通过详细研究两个重要的例子，我们将勾勒出分离三元组$(W,p,ψ)$的方法的范围。值得注意的是，"分离三元组"和"对子元的限制"方法足以证明所有维数为$\le 7$的指数可解李群都是本原$\ast$-正则的.
英文摘要：
Let $G$ be an exponential solvable Lie group. By definition $G$ is $\ast$-regular if $ker_{L^1(G)}π$ is dense in $ker_{C^\ast(G)}π$ for all unitary representations $π$ of $G$. Boidol characterized the $\ast$-regular exponential Lie groups by a purely algebraic condition. In this article we will focus on non-$\ast$-regular groups. We say that $G$ is primitive $\ast$-regular if the above density condition is satisfied for all irreducible representations. Our goal is to develop appropriate tools to verify this weaker property. To this end we will introduce Duflo pairs $(W,p)$ and central Fourier multipliers $ψ$ on the stabilizer $M=G_fN$ of representations $π=\mcK(f)$ in general position. Using Littlewood-Paley theory we will derive some results on multiplier operators $T_ψ$ which might be of independent interest. The scope of our method of separating triples $(W,p,ψ)$ will be sketched by studying two significant examples in detail. It should be noticed that the methods 'separating triples' and 'restriction to subquotients' suffice to prove that all exponential solvable Lie groups of dimension $\le 7$ are primitive $\ast$-regular.
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