Closed orbits for actions of maximal tori on homogeneous spaces
The action of a torus \(T\) (containing a maximal \(R\)-split torus) on a homogeneous space is studied. One considers a group \(G\) defined over the set \(Q\) of rational numbers and then the homogeneous space \(G/\Gamma \) is constructed. It is proven that the closed orbits for the action of \(T\) on \(G/\Gamma \) admit a simple algebraic description. When \(G\) is reductive, it is shown that an orbit \(Tx\Gamma \) is closed if and only if \(x^{-1}Tx\) is a product of a compact torus and a torus defined over \(Q.\) In addition, it is established that such an orbit is divergent if and only if the maximal \(R\)-split subtorus of \(x^{-1}Tx\) is defined over \(Q\) and \(Q\)-split. It is concluded that all closed orbits are ``standard'', that is they correspond to \(Q\)-subtori in \(G\). Some illustrative examples are also given and open questions for further researches are formulated.
scientific article; zbMATH DE number 2068476
scientific article