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

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

2011-06-14

arxiv.org/abs/1106.2623v1

Mathematics

Dynamical Systems

52 pages

Scientific paper

Let $N$ be a connected and simply connected nilpotent Lie group, $\Lambda$ a lattice in $N$, and $X=N/\Lambda$ 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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