Random products of automorphisms of Heisenberg nilmanifolds and Weil's representation
Résumé
For $n \geq 1$, let $H$ be the $(2n+1)$-dimensional real Heisenberg group, and let $\wedge$ be a lattice in $H$. Let $\Gamma$ be a group of automorphisms of the corresponding nilmanifold $\wedge\setminus H$ and $U$ the associated unitary representation of $\Gamma$ on $L^{2}(\wedge\setminus H)$. Denote by $T$ the maximal torus factor associated to $\wedge\setminus H$. Using Weil's representation (also known as the metaplectic representation), we show that a dense set of matrix coefficients of the restriction of $U$ to the orthogonal complement of $L^{2}(T)$ in $L^{2}(\wedge\setminus H)$ belong to $\ell^{4n+2+\varepsilon} (\Gamma)$ for every $\varepsilon> 0$. We give the following application to random walks on $\wedge\setminus H$ defined by a probability measure $\mu$ on Aut$(\wedge\setminus H)$. Denoting by $\Gamma$ the subgroup of Aut$(\wedge\setminus H)$ generated by the support of $\mu$ and by $U^{0}$ and $V^{0}$ the restrictions of $U$ respectively to the subspaces of $L^{2}(\wedge\setminus H)$ and $L^{2}(T)$ with zero mean, we prove the following inequality: $$\|U^{0}(\mu)\| \leq max \{\|V^{0}\mu\|, \|\lambda_{\Gamma} (\mu)\|^{1/(2n+2)}\},$$ where $\ambda_{\Gamma}$ is the left regular representation of $\Gamma$ on $\ell^{2}(\Gamma)$. In particular, the action of $\Gamma$ on $\wedge\setminus H$ has a spectral gap if and only if the corresponding action of $\Gamma$ on $T$ has a spectral gap.
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