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Let $N$ be a connected and simply connected nilpotent Lie group, $\Lambda$ alattice in $N$, and $X=N/\Lambda$ the corresponding nilmanifold. Let $Aff(X)$be the group of affine transformations of $X$. We characterize the countablesubgroups $H$ of $Aff(X)$ for which the action of $H$ on $X$ has a spectralgap, that is, such that the associated unitary representation $U$ of $H$ on thespace of functions from $L^2(X)$ with zero mean does not weakly contain thetrivial 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 ifthere exists no proper $H$-invariant subtorus $S$ of $T$ such that theprojection of $H$ on $Aut (T/S)$ has an abelian subgroup of finite index. Wefirst establish the result in the case where $X$ is a torus. In the case of ageneral nilmanifold, we study the asymptotic behaviour of matrix coefficientsof $U$ using decay properties of metaplectic representations of symplecticgroups. 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 stronglymixing).

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