Let G be a real algebraic group defined over ℚ, let Γ be an arithmetic subgroup, and let T be any torus containing a maximal ℝ-split torus. We prove that the closed orbits for the action of T on G/Γ admit a simple algebraic description. In particular, we show that if G is reductive, an orbit TxΓ is closed if and only if x-1 Tx is a product of a compact torus and a torus defined over ℚ, and it is divergent if and only if the maximal ℝ-split subtorus x-1 Tx is defined over ℚ and ℚ-split. Our analysis also yields the following: there is a compact K ⊂ G/Γ which intersects every T-orbit; if rankℚ G < rankℝ G, there are no divergent orbits for T.
ASJC Scopus subject areas
- General Mathematics