Let \(G/K\) be a symmetric space of non-compact type. One can identify \(T(G/K)\) with \(T^{\ast}(G/K)\) by using the Riemannian metric of \(G/K\), and \(T_{eK}(G/K)\) with \(\mathfrak{p}\) through the corresponding Cartan decomposition \(\mathfrak{g}=\mathfrak{k}+\mathfrak{p}\). Let \(\mathcal{A}\) be the algebra of \(G\)-invariant smooth functions on \(T(G/K)\cong T^{\ast}(G/K)\) which restrict to polynomials on \(\mathfrak{p}\). The author defines a natural action \(\mathcal{A}\) on \(T(G/K)\cong T^{\ast}(G/K)\) by suitable Hamiltonian flows. This action has some remarkable properties, which follow from the main theorem 8.3 (Ergodic Arnold-Liouville theorem for locally symmetric spaces) of the paper. For instance, this theorem states that the joint level set of \(\mathcal{A}\) are precisely the \(G\)-orbits in \(T(G/K)\). If \(\Gamma\) is a lattice in \(G\), then the \(\mathcal{A}\)-action factors to an action on \(\Gamma\backslash T(G/K)\cong T(\Gamma\backslash G/K)\), and by the above theorem, the restriction of this \(\mathcal{A}\)-action to \(\Gamma\backslash O\), where \(O\) is a generic \(G\)-orbit in \(T(G/K)\), is ergodic.
For the entire collection see [Zbl 1089.53002].