arXiv 1106.26232011-06-14math.DS

关于紧致流形的仿射变换群的spectral theory

On the spectral theory of groups of affine transformations of compact nilmanifolds

Bachir Bekka, Yves Guivarc'h

备注：: 52 pages
发表信息：: Ann. Sci. Ecole Normale Superieure 48 (2015), 607-645

摘要：

设$N$是连通和单连通的幂零李群，$Λ$是$N$中的格，$X=N/Λ$是相应的幂零流形.令$Aff(X)$为$X$的仿射变换群。我们刻画了$Aff(X)$的可数子群$H$，其中$H$对$X$的作用具有谱隙，即使得$H$在具有零均值的$L^2(X)$的函数空间上的相关酉表示$U$不弱包含平凡表示.用$T$表示与$X$相关的最大环面因子。证明了$H$对$X$的作用有谱隙当且仅当不存在$T$的真$H$不变子环面$S$，使得$H$在$Aut (T/S)$上的投影有有限指数的阿贝尔子群.我们首先在$X$是环面的情况下建立结果。在一般零流形的情况下，我们利用辛群的元辛表示的衰减性质研究了$U$矩阵系数的渐进行为。结果表明，$Aff(X)$子群的谱间隙的存在等价于K.Schmidt意义下的强遍历性.此外，我们还证明了$H$对$X$的作用是遍历的（或强混合的）当且仅当相应的$H$对$T$的作用是遍历的（或强混合的）。

英文摘要：

Let $N$ be a connected and simply connected nilpotent Lie group, $Λ$ a lattice in $N$, and $X=N/Λ$ the corresponding nilmanifold. Let $Aff(X)$ be the group of affine transformations of $X$. We characterize the countable subgroups $H$ of $Aff(X)$ for which the action of $H$ on $X$ has a spectral gap, that is, such that the associated unitary representation $U$ of $H$ on the space of functions from $L^2(X)$ with zero mean does not weakly contain the trivial representation. Denote by $T$ the maximal torus factor associated to $X$. We show that the action of $H$ on $X$ has a spectral gap if and only if there exists no proper $H$-invariant subtorus $S$ of $T$ such that the projection of $H$ on $Aut (T/S)$ has an abelian subgroup of finite index. We first establish the result in the case where $X$ is a torus. In the case of a general nilmanifold, we study the asymptotic behaviour of matrix coefficients of $U$ using decay properties of metaplectic representations of symplectic groups. The result shows that the existence of a spectral gap for subgroups of $Aff(X)$ is equivalent to strong ergodicity in the sense of K.Schmidt. Moreover, we show that the action of $H$ on $X$ is ergodic (or strongly mixing) if and only if the corresponding action of $H$ on $T$ is ergodic (or strongly mixing).

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