TY - JOUR

T1 - Closed orbits for actions of maximal tori on homogeneous spaces

AU - Tomanov, George

AU - Weiss, Barak

PY - 2003/8/15

Y1 - 2003/8/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0141756161&partnerID=8YFLogxK

U2 - 10.1215/S0012-7094-03-11926-2

DO - 10.1215/S0012-7094-03-11926-2

M3 - Article

AN - SCOPUS:0141756161

VL - 119

SP - 367

EP - 392

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 2

ER -