Quasi-regular representations of two-step nilmanifolds
Let \(G\) be a simply connected and connected nilpotent Lie group with \(\Gamma \leq G\) a uniform lattice in \(G\), then we call the quotient space \(G / \Gamma\) a \textit{nilmanifold}, where \(\Gamma\) acts by right translations on \(G\). Nilmanifolds play an important role in both geometry and dynamical systems. For example it was shown in [\textit{M. Gromov}, Publ. Math., Inst. Hautes Étud. Sci. 53, 53--78 (1981; Zbl 0474.20018)] that these are up to finite cover the only manifolds admitting expanding maps (and it is conjectured that the same holds for Anosov diffeomorphisms). Note that the Lie group \(G\) acts on the manifold \(G / \Gamma\) by left translations and hence gives rise to a unitary representation of \(G\) on \(L^2(G/\Gamma)\), called the \textit{quasi-regular representation} of \(G / \Gamma\). The paper under review studies explicit unitary equivalences between quasi-regular representations of nilmanifolds in the special case where the Lie group \(G\) is \(2\)-step nilpotent. In [\textit{R. Gornet}, J. Funct. Anal. 119, No. 1, 121--137 (1994; Zbl 0830.22007)] it was shown that two nilmanifolds \(G / \Gamma_1\) and \(G/ \Gamma_2\) have unitarily equivalent quasi-regular representations if and only if there exists a \(\Gamma_1\)-equivalence \(\Phi\) of \(G\) such that \(\Phi(\Gamma_1) = \Gamma_2\). These \(\Gamma_1\)-equivalences were introduced in the same paper as automorphisms of \(G\) satisfying an extra condition on the elements of \(\Gamma_1\). The main result of the paper, which is Theorem 4.6, gives an explicit form for the intertwining operator depending on the \(\Gamma_1\)-equivalence \(\Phi\). For the entire collection see [Zbl 1392.42001].
scientific article; zbMATH DE number 6986093