## Abstract

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_{ℚ} G < rank_{ℝ} G, there are no divergent orbits for T.

Original language | English |
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Pages (from-to) | 367-392 |

Number of pages | 26 |

Journal | Duke Mathematical Journal |

Volume | 119 |

Issue number | 2 |

DOIs | |

State | Published - 15 Aug 2003 |

## ASJC Scopus subject areas

- General Mathematics