TY - JOUR

T1 - On actions of epimorphic subgroups on homogeneous spaces

AU - Shah, Nimish A.

AU - Weiss, Barak

PY - 2000/4

Y1 - 2000/4

N2 - For an inclusion F < G < L of connected real algebraic groups such that F is epimorphic in G, we show that any closed F-invariant subset of L/Λ is G-invariant, where Λ is a lattice in L. This is a topological analogue of a result due to S. Mozes, that any finite F-invariant measure on L/Λ is G-invariant. This result is established by proving the following result. If in addition G is generated by unipotent elements, then there exists a ∈ F such that the following holds. Let U ⊂ F be the subgroup generated by all unipotent elements of F, x ∈ L/Λ, and λ and μ denote the Haar probability measures on the homogeneous spaces Ux and Gx, respectively (cf. Ratner's theorem). Then anλ → μ weakly as n → ∞. We also give an algebraic characterization of algebraic subgroups F < SLn(ℝ) for which all orbit closures on SLn(ℝ)/SLn(ℤ) are finite-volume almost homogeneous, namely the smallest observable subgroup of SLn(ℝ) containing F should have no non-trivial algebraic characters defined over ℝ.

AB - For an inclusion F < G < L of connected real algebraic groups such that F is epimorphic in G, we show that any closed F-invariant subset of L/Λ is G-invariant, where Λ is a lattice in L. This is a topological analogue of a result due to S. Mozes, that any finite F-invariant measure on L/Λ is G-invariant. This result is established by proving the following result. If in addition G is generated by unipotent elements, then there exists a ∈ F such that the following holds. Let U ⊂ F be the subgroup generated by all unipotent elements of F, x ∈ L/Λ, and λ and μ denote the Haar probability measures on the homogeneous spaces Ux and Gx, respectively (cf. Ratner's theorem). Then anλ → μ weakly as n → ∞. We also give an algebraic characterization of algebraic subgroups F < SLn(ℝ) for which all orbit closures on SLn(ℝ)/SLn(ℤ) are finite-volume almost homogeneous, namely the smallest observable subgroup of SLn(ℝ) containing F should have no non-trivial algebraic characters defined over ℝ.

UR - http://www.scopus.com/inward/record.url?scp=0034421403&partnerID=8YFLogxK

U2 - 10.1017/S0143385700000298

DO - 10.1017/S0143385700000298

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:0034421403

SN - 0143-3857

VL - 20

SP - 567

EP - 592

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 2

ER -