Closed orbits for actions of maximal tori on homogeneous spaces

George Tomanov, Barak Weiss

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


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 rank G < rank G, there are no divergent orbits for T.

Original languageEnglish
Pages (from-to)367-392
Number of pages26
JournalDuke Mathematical Journal
Issue number2
StatePublished - 15 Aug 2003

ASJC Scopus subject areas

  • General Mathematics


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